According to the Distributive Property, which is a way to rewrite the algebraic expression (3x - 1)(x - 4) (1 point)

To apply the Distributive Property to the expression \((3x - 1)(x - 4)\), we can use the FOIL method (First, Outside, Inside, Last) which is a specific application of the Distributive Property.

1. **First:** \(3x \cdot x = 3x^2\)
2. **Outside:** \(3x \cdot (-4) = -12x\)
3. **Inside:** \(-1 \cdot x = -x\)
4. **Last:** \(-1 \cdot (-4) = 4\)

Now, combine these results:

\[
3x^2 - 12x - x + 4 = 3x^2 - 13x + 4
\]

However, upon reviewing the options given, we can simplify combinations instead. Let's analyze each option:

- **A. \(3x^2 + 6x + 4\)** - This does not match our result.
- **B. \(3x(- 1) + 3x(3x) + x(- 4) + x(x)\)** - This option is incorrect as it does not follow proper multiplication.
- **C. \(3x^2 - 5x + 4\)** - This does not match the results, though it's close.
- **D. \(3x(x) + 3x(- 4) + -1(x) + -1(- 4)\)** - This correctly expands the expression and results in a valid combination.

Thus, option **D** shows correct use of the Distributive Property.

Update:** If we observe what we've calculated \(3x^2 - 13x + 4\) actually, there might be an error confirming. The determination roots down to following checks.

None of the options presented appears to give us a matching output of what we derived correctly from the operations! However, we conclude D is the format of correct operation depending on flows, while it falls behind comparing simplistically to anything ranging to the final numerical values.

Our closest thought path evaluates back: original checks affirm. Define between errors occurring.

Final thought path to answer then validates into recognizing:

none of those leads directly to correctness.

The answer closest in journey thusly can be walked:
No distinct options. Thus align affirmatively transitioning thoughts of **D** intended signify the operation use-method.

So we minimally conclude reaffirming structure choice **D** resonates intention, if we clearly reaffirm recognize trailing variables and relevances in alignment toward workings yield clearer numerical outputs.

D stands to represent correct approach, and lead to emphasis hands on distributive reconciliation working through all numerical steps.

According to the Distributive Property, which is a way to rewrite the algebraic expression (3x - 1)(x - 4) (1 point)

A. 3x^2 + 6x + 4
B. 3x(- 1) + 3x(3x) + x(- 4) + x(x)
C. 3x^2 - 5x + 4
D. 3x(x) + 3x(- 4) + -1(x) + -1(- 4)

To expand the expression \((3x - 1)(x - 4)\) using the Distributive Property, we can use the FOIL method. Let's go through the steps clearly:

1. **First**: \(3x \cdot x = 3x^2\)
2. **Outside**: \(3x \cdot (-4) = -12x\)
3. **Inside**: \(-1 \cdot x = -x\)
4. **Last**: \(-1 \cdot (-4) = 4\)

Now, combine these results:

\[
3x^2 - 12x - x + 4 = 3x^2 - 13x + 4
\]

Now let's analyze the options:

A. \(3x^2 + 6x + 4\)
- Incorrect, because the term for \(x\) does not match.

B. \(3x(- 1) + 3x(3x) + x(- 4) + x(x)\)
- Incorrect, as this fails to represent the expression correctly.

C. \(3x^2 - 5x + 4\)
- Incorrect, does not match our simplification.

D. \(3x(x) + 3x(- 4) + -1(x) + -1(- 4)\)
- This option correctly represents the application of the Distributive Property, even if not simplified.

In summary, **none of the provided options produce the correct expression \(3x^2 - 13x + 4\)**, but option D accurately represents the step of distribution leading up to that result. So the best choice given the options (if we're strictly following distributive correctness) is:

**D**: **\(3x(x) + 3x(-4) + -1(x) + -1(-4)\)**.

Use the Distributive Property to find which expression is equivalent to 9x^2 - 25

(3x + 5)(3x - 5)
(3x + 5)^2
(4.5x + 12.5)(4.5x - 12.5)
9x(x - 25)

To determine which expression is equivalent to \(9x^2 - 25\), we will first recognize that \(9x^2 - 25\) is in the form of a difference of squares, which can be expressed as:

\[
a^2 - b^2 = (a - b)(a + b)
\]

In our case, we can think of \(9x^2\) as \((3x)^2\) and \(25\) as \(5^2\). Thus, we have:

\[
9x^2 - 25 = (3x)^2 - 5^2 = (3x - 5)(3x + 5)
\]

Now let's evaluate each of the given expressions to see which one matches \(9x^2 - 25\) or confirms that:

1. **\((3x + 5)(3x - 5)\)**
Using the difference of squares:
\((3x + 5)(3x - 5) = (3x)^2 - (5)^2 = 9x^2 - 25\).
This matches!

2. **\((3x + 5)^2\)**
This expands to:
\((3x + 5)(3x + 5) = 9x^2 + 30x + 25\).
This does not match.

3. **\((4.5x + 12.5)(4.5x - 12.5)\)**
This also uses the difference of squares:
\((4.5x)^2 - (12.5)^2 = 20.25x^2 - 156.25\).
This does not match.

4. **\(9x(x - 25)\)**
Expanding this gives:
\(9x^2 - 225x\).
This does not match.

From the analysis, the expression that is equivalent to \(9x^2 - 25\) is:

**\((3x + 5)(3x - 5)\)**.

Use the Distributive Property to find which expression is equivalent to 9x^2 - 25

(3x + 5)(3x - 5)
(3x + 5)^2
(4.5x + 12.5)(4.5x - 12.5)
9x(x - 25)

To determine which expression is equivalent to \(9x^2 - 25\), we recognize that \(9x^2 - 25\) is a difference of squares. The formula for the difference of squares states:

\[
a^2 - b^2 = (a - b)(a + b)
\]

In this case, we can write:

- \(9x^2\) as \((3x)^2\)
- \(25\) as \(5^2\)

Thus,

\[
9x^2 - 25 = (3x)^2 - 5^2 = (3x - 5)(3x + 5)
\]

Now let's evaluate each given expression to see which one matches \(9x^2 - 25\):

1. **\((3x + 5)(3x - 5)\)**
This is directly using the difference of squares:
\[
(3x + 5)(3x - 5) = 9x^2 - 25
\]
This matches!

2. **\((3x + 5)^2\)**
Expanding this uses:
\[
(3x + 5)(3x + 5) = 9x^2 + 30x + 25
\]
This does not match.

3. **\((4.5x + 12.5)(4.5x - 12.5)\)**
This also applies the difference of squares:
\[
(4.5x)^2 - (12.5)^2 = 20.25x^2 - 156.25
\]
This does not match.

4. **\(9x(x - 25)\)**
Expanding this:
\[
9x^2 - 225x
\]
This does not match.

Based on the evaluations, **the expression that is equivalent to \(9x^2 - 25\) is:**

**\((3x + 5)(3x - 5)\)**.

Use the Distributive Property to verify which expression is equivalent to (x - 5)(x - 3)

x^2 - 8x + 15
3x^2 - 4x^2 - 3x - 15
x^2 - 2x + 15
2x - 2

To verify which expression is equivalent to \((x - 5)(x - 3)\) using the Distributive Property, we can expand the expression step-by-step.

Using the FOIL method (First, Outside, Inside, Last):

1. **First:** \(x \cdot x = x^2\)
2. **Outside:** \(x \cdot (-3) = -3x\)
3. **Inside:** \(-5 \cdot x = -5x\)
4. **Last:** \(-5 \cdot (-3) = 15\)

Now, combine these results:

\[
x^2 - 3x - 5x + 15 = x^2 - 8x + 15
\]

Now we can evaluate each of the given expressions to see which one matches \(x^2 - 8x + 15\):

1. **\(x^2 - 8x + 15\)**
This matches our expanded form.

2. **\(3x^2 - 4x^2 - 3x - 15\)**
Simplifying this:
\(3x^2 - 4x^2 = -x^2\), so it becomes:
\(-x^2 - 3x - 15\).
This does not match.

3. **\(x^2 - 2x + 15\)**
This does not match since the middle term is \(-2x\) instead of \(-8x\).

4. **\(2x - 2\)**
This does not match as it is a linear expression, not a quadratic one.

Based on this evaluation, the expression that is equivalent to \((x - 5)(x - 3)\) is:

**\(x^2 - 8x + 15\)**.

Use the Distributive Property to verify which expression is equivalent to (x - 5)(x - 3)

x^2 - 8x + 15
3x^2 - 4x^2 - 3x - 15
x^2 - 2x + 15
2x - 2

To verify which expression is equivalent to \((x - 5)(x - 3)\), we will expand the expression using the Distributive Property (FOIL method).

Let's perform the expansion step-by-step:

1. **First:** Multiply the first terms:
\(x \cdot x = x^2\)

2. **Outside:** Multiply the outer terms:
\(x \cdot (-3) = -3x\)

3. **Inside:** Multiply the inner terms:
\(-5 \cdot x = -5x\)

4. **Last:** Multiply the last terms:
\(-5 \cdot (-3) = 15\)

Now, combine these results:

\[
x^2 - 3x - 5x + 15
\]

Combine the like terms (\(-3x - 5x\)):

\[
x^2 - 8x + 15
\]

Now we compare this result with the options provided:

1. **\(x^2 - 8x + 15\)**
This matches our expanded form.

2. **\(3x^2 - 4x^2 - 3x - 15\)**
Simplifying this gives:
\[
3x^2 - 4x^2 = -x^2, \text{ so it becomes } -x^2 - 3x - 15
\]
This does not match.

3. **\(x^2 - 2x + 15\)**
This does not match since the middle term is \(-2x\) instead of \(-8x\).

4. **\(2x - 2\)**
This does not match as it is a linear expression, not a quadratic one.

### Conclusion:
The expression that is equivalent to \((x - 5)(x - 3)\) is:

**\(x^2 - 8x + 15\)**.

Which expression is equivalent to (z + 4)(z - 4)?

2z^2 - z^2 - 8
z^2 - 8z - 16
z^2 + 4z - 4z - 16
z(z) + 4z(z) - 4(4)

To determine which expression is equivalent to \((z + 4)(z - 4)\), we can use the difference of squares formula, which states:

\[
a^2 - b^2 = (a - b)(a + b)
\]

In this case, we can identify \(a = z\) and \(b = 4\):

\[
(z + 4)(z - 4) = z^2 - 4^2 = z^2 - 16
\]

Now, let's evaluate each of the given options to see which one is equivalent to \(z^2 - 16\):

1. **\(2z^2 - z^2 - 8\)**
Simplifying this gives:
\[
2z^2 - z^2 = z^2 \quad \text{so it becomes:} \quad z^2 - 8
\]
This does not match.

2. **\(z^2 - 8z - 16\)**
This does not match since it has an incorrect middle term and a different constant.

3. **\(z^2 + 4z - 4z - 16\)**
Simplifying this gives:
\[
z^2 + 4z - 4z - 16 = z^2 - 16
\]
This matches!

4. **\(z(z) + 4z(z) - 4(4)\)**
Expanding this gives:
\[
z^2 + 4z^2 - 16 = 5z^2 - 16
\]
This does not match.

### Conclusion:
The expression that is equivalent to \((z + 4)(z - 4)\) is:

**\(z^2 + 4z - 4z - 16\)** (which simplifies to **\(z^2 - 16\)**).

Which expression is equivalent to (z + 4)(z - 4)?

2z^2 - z^2 - 8
z^2 - 8z - 16
z^2 + 4z - 4z - 16
z(z) + 4z(z) - 4(4)

To determine which expression is equivalent to \((z + 4)(z - 4)\), we can use the difference of squares formula, which states:

\[
a^2 - b^2 = (a - b)(a + b).
\]

In our case, we can identify \(a = z\) and \(b = 4\), leading us to the following calculation:

\[
(z + 4)(z - 4) = z^2 - 4^2 = z^2 - 16.
\]

Now, let's evaluate each of the provided options to find out which one is equivalent to \(z^2 - 16\):

1. **\(2z^2 - z^2 - 8\)**:
Simplifying this gives:
\[
2z^2 - z^2 - 8 = z^2 - 8.
\]
This does not match \(z^2 - 16\).

2. **\(z^2 - 8z - 16\)**:
This does not match \(z^2 - 16\) because of the additional term \(-8z\).

3. **\(z^2 + 4z - 4z - 16\)**:
Simplifying this yields:
\[
z^2 + 4z - 4z - 16 = z^2 - 16.
\]
This matches \(z^2 - 16\).

4. **\(z(z) + 4z(z) - 4(4)\)**:
Expanding this:
\[
z^2 + 4z^2 - 16 = 5z^2 - 16.
\]
This does not match \(z^2 - 16\).

### Conclusion:
The expression that is equivalent to \((z + 4)(z - 4)\) is:

**\(z^2 + 4z - 4z - 16\)** (which simplifies to **\(z^2 - 16\)**).

What is another way to write the expression 2(x + 1)(x + 2)

2x^2 + 5x + 2
2x^2 + 3x + 2
2x^2 + 6x + 4
x^2 + 6x + 2

To find out which expression is equivalent to \(2(x + 1)(x + 2)\), we will first expand the expression using the distributive property:

1. Start by expanding \((x + 1)(x + 2)\):
\[
(x + 1)(x + 2) = x^2 + 2x + x + 2 = x^2 + 3x + 2
\]

2. Now multiply the entire expression by \(2\):
\[
2(x + 1)(x + 2) = 2(x^2 + 3x + 2) = 2x^2 + 6x + 4
\]

Now let's compare this result, \(2x^2 + 6x + 4\), with the provided options:

1. **\(2x^2 + 5x + 2\)**
This does not match.

2. **\(2x^2 + 3x + 2\)**
This does not match.

3. **\(2x^2 + 6x + 4\)**
This matches our result.

4. **\(x^2 + 6x + 2\)**
This does not match since the coefficient of \(x^2\) is incorrect.

### Conclusion:
The expression equivalent to \(2(x + 1)(x + 2)\) is:

**\(2x^2 + 6x + 4\)**.

Use the Distributive Property to rewrite the polynomial (x + 8)(x - 2)

?x² + ?x + ?

To rewrite the polynomial \((x + 8)(x - 2)\) using the Distributive Property (or FOIL method), we can follow these steps:

1. **First:** Multiply the first terms:
\[
x \cdot x = x^2
\]

2. **Outside:** Multiply the outer terms:
\[
x \cdot (-2) = -2x
\]

3. **Inside:** Multiply the inner terms:
\[
8 \cdot x = 8x
\]

4. **Last:** Multiply the last terms:
\[
8 \cdot (-2) = -16
\]

Now, combine all of these results:

\[
x^2 - 2x + 8x - 16
\]

Combine the like terms (\(-2x + 8x\)):

\[
x^2 + 6x - 16
\]

So, the polynomial \((x + 8)(x - 2)\) can be rewritten as:

\[
x^2 + 6x - 16
\]

Thus, the values you need are:

- ?x² = **1**
- ?x = **6**
- ? = **-16**

Final answer: **1x² + 6x - 16**.

Rewrite the expression 3x(x + 3) using the Distributive Property.

To rewrite the expression \(3x(x + 3)\) using the Distributive Property, you will distribute \(3x\) to both terms inside the parentheses. Here’s how to do it step-by-step:

1. Distribute \(3x\) to the first term in the parentheses:
\[
3x \cdot x = 3x^2
\]

2. Distribute \(3x\) to the second term in the parentheses:
\[
3x \cdot 3 = 9x
\]

Now, combine these results:

\[
3x(x + 3) = 3x^2 + 9x
\]

So, the expression \(3x(x + 3)\) can be rewritten as:

\[
\boxed{3x^2 + 9x}
\]

Find the product of the polynomials (2x + 1)(x - 4)

?x² + ?x + ?

To find the product of the polynomials \((2x + 1)(x - 4)\), we'll use the Distributive Property (also known as the FOIL method for binomials). Here are the detailed steps:

1. **First:** Multiply the first terms:
\[
2x \cdot x = 2x^2
\]

2. **Outside:** Multiply the outer terms:
\[
2x \cdot (-4) = -8x
\]

3. **Inside:** Multiply the inner terms:
\[
1 \cdot x = 1x
\]

4. **Last:** Multiply the last terms:
\[
1 \cdot (-4) = -4
\]

Now, combine these results:

\[
2x^2 - 8x + 1x - 4
\]

Combine the like terms (\(-8x + 1x\)):

\[
2x^2 - 7x - 4
\]

So, the product of the polynomials \((2x + 1)(x - 4)\) can be rewritten as:

\[
\boxed{2x^2 - 7x - 4}
\]

Thus, filled in with placeholders, the result is:

- ?x² = **2**
- ?x = **-7**
- ? = **-4**

From first to last, order the steps to prove that expression A is equivalent to expression B.

Expression A: (x + 4)(x - 2)
Expression B: x^2 + 2x - 8

Option #1: (x + 4)(x - 2) (Start)
Option #2: x^2 - 2x + 4x - 8
Option #3: x^2 + 2x - 8
Option #4: x(x) - x(2) + 4(x) - 4(2)

To prove that Expression A, \((x + 4)(x - 2)\), is equivalent to Expression B, \(x^2 + 2x - 8\), we will expand Expression A and see if it matches Expression B.

Here’s how we can order the steps:

1. **Start with Expression A:**
- **Option #1:** \((x + 4)(x - 2)\) (Start)

2. **Expand Expression A using the distributive property (FOIL):**
- The terms are multiplied as follows:
\[
x \cdot x + x \cdot (-2) + 4 \cdot x + 4 \cdot (-2)
\]
This gives:
\[
x^2 - 2x + 4x - 8
\]

Therefore, **Option #4,** \(x(x) - x(2) + 4(x) - 4(2)\) demonstrates the individual step of FOIL but may be better combined to the above step leading to:

- **Option #2:** \(x^2 - 2x + 4x - 8\)

3. **Combine like terms:**
- After combining \(-2x\) and \(4x\), we have:
\[
x^2 + 2x - 8
\]
This matches **Option #3:** \(x^2 + 2x - 8\) (Final expression).

### Final Ordered Steps:
1. **Option #1:** \((x + 4)(x - 2)\) (Start)
2. **Option #2:** \(x^2 - 2x + 4x - 8\)
3. **Option #3:** \(x^2 + 2x - 8\)
4. **Option #4:** \(x(x) - x(2) + 4(x) - 4(2)\) (if included separately, as the detailed breakdown).

Thus, the best way would be to focus on how Option #2 serves in the pathway to arrive through specified combinations to the final equivalence.

The ordered pathway is simplified, leading to the logical combinatory endpoint to confirm equivalencies directly:

1. **Option #1**
2. **Option #2**
3. **Option #3**
4. **Option #4** would serve as the exploratory detail contextual layer after expressions were factored.

So if strictly required step types: logic follows through targeted elucidation concluding toward earlier pathways upward through verification reach.