Use the Distributive Property to rewrite the polynomial (x+8)(x−2) .(2 points)

To use the Distributive Property to rewrite the polynomial \( (x+8)(x-2) \), we will distribute each term in the first parenthesis \( (x + 8) \) to each term in the second parenthesis \( (x - 2) \).

Here’s how it works step by step:

1. Distribute \( x \) in \( (x + 8) \):
\[
x \cdot x = x^2
\]
\[
x \cdot (-2) = -2x
\]

2. Distribute \( 8 \) in \( (x + 8) \):
\[
8 \cdot x = 8x
\]
\[
8 \cdot (-2) = -16
\]

3. Now, combine all the results:
\[
x^2 - 2x + 8x - 16
\]

4. Combine the like terms \(-2x\) and \(8x\):
\[
x^2 + 6x - 16
\]

Thus, the polynomial \( (x+8)(x-2) \) can be rewritten as:
\[
x^2 + 6x - 16
\]

Rewrite the expression 3x(x + 3) using the Distributive Property.(1 point)

To rewrite the expression \( 3x(x + 3) \) using the Distributive Property, you will distribute \( 3x \) to each term inside the parentheses \( (x + 3) \).

Here are the steps:

1. Distribute \( 3x \) to \( x \):
\[
3x \cdot x = 3x^2
\]

2. Distribute \( 3x \) to \( 3 \):
\[
3x \cdot 3 = 9x
\]

3. Combine the results:
\[
3x(x + 3) = 3x^2 + 9x
\]

So, the expression \( 3x(x + 3) \) can be rewritten as:
\[
3x^2 + 9x
\]

Find the product of the polynomials (2x+1)(x−4) .(2 points)

x2+
x+
Remaining Attempts : 3

To find the product of the polynomials \( (2x+1)(x-4) \), we will use the Distributive Property (also known as the FOIL method for binomials).

Here's how to do it step by step:

1. **Distribute \(2x\)**:
- \(2x \cdot x = 2x^2\)
- \(2x \cdot (-4) = -8x\)

2. **Distribute \(1\)**:
- \(1 \cdot x = x\)
- \(1 \cdot (-4) = -4\)

3. **Combine all the results**:
\[
2x^2 - 8x + x - 4
\]

4. **Combine the like terms \(-8x\) and \(x\)**:
\[
2x^2 - 7x - 4
\]

Thus, the product of the polynomials \( (2x+1)(x-4) \) is:
\[
2x^2 - 7x - 4
\]

So, if you wish to fill in the blank:
- \(x^2\) coefficient: **2**
- \(x\) coefficient: **-7**
- Constant term: **-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: x2+2x−8



Option #1: (x+4)(x−2) (Start)

Option #2: x2−2x+4x−8

Option #3: x2+2x−8

Option #4: x(x)−x(2)+4(x)−4(2)

(2 points)
First: Option #

Second: Option #

Third: Option #

Fourth: Option #

To prove that Expression A \((x + 4)(x - 2)\) is equivalent to Expression B \(x^2 + 2x - 8\), we need to follow the steps of using the Distributive Property (or FOIL method). Here’s the correct order:

1. **First: Option #1**: \((x + 4)(x - 2)\) (Start)

2. **Second: Option #4**: \(x(x) - x(2) + 4(x) - 4(2)\) (Distributing each term)

3. **Third: Option #2**: \(x^2 - 2x + 4x - 8\) (Combine the products)

4. **Fourth: Option #3**: \(x^2 + 2x - 8\) (Combine like terms)

So, the ordering is:
- **First: Option #1**
- **Second: Option #4**
- **Third: Option #2**
- **Fourth: Option #3**

Identify two expressions that will be equivalent to 2x2−8x−10 when distributed. Enter the option with the lesser number first.

Option #1: 2x(x−4x−5)

Option #2: (2x+2)(x−5)

Option #3: 2x(x−5)+2(x−5)

Option #4: (x−5)(x−3)

Option #5: −6x−10

(1 point)
Option #
and Option #

To identify which expressions are equivalent to \(2x^2 - 8x - 10\) when distributed, let's simplify each of the given options.

### Option #1: \(2x(x-4x-5)\)
This option looks like it contains a typo (two variables after \(4\)). Assuming it means \(2x(x - 4)(x - 5)\):
- **Distribution**: \(2x[x - (4x + 5)] = 2x(x - 4x - 5)\) is not correct in structure as it overcomplicates.

### Option #2: \((2x + 2)(x - 5)\)
- **Distribution**:
\[
(2x + 2)(x - 5) = 2x \cdot x + 2x \cdot (-5) + 2 \cdot x + 2 \cdot (-5)
\]
\[
= 2x^2 - 10x + 2x - 10 = 2x^2 - 8x - 10
\]
- This expression is equivalent to \(2x^2 - 8x - 10\).

### Option #3: \(2x(x - 5) + 2(x - 5)\)
- **Distribution**:
\[
2x(x - 5) + 2(x - 5) = 2x^2 - 10x + 2x - 10
\]
\[
= 2x^2 - 8x - 10
\]
- This expression is equivalent to \(2x^2 - 8x - 10\).

### Option #4: \((x - 5)(x - 3)\)
- **Distribution**:
\[
(x - 5)(x - 3) = x^2 - 3x - 5x + 15 = x^2 - 8x + 15
\]
- This is not equivalent to \(2x^2 - 8x - 10\).

### Option #5: \(-6x - 10\)
- This is not a polynomial in \(x^2\), hence not equivalent.

### Summary:
The equivalent expressions are:
- **Option #2**: \((2x + 2)(x - 5)\)
- **Option #3**: \(2x(x - 5) + 2(x - 5)\)

In order of lesser number first:
- **Option #2**
- **Option #3**

So the answer is:
**Option #2 and Option #3**.