Question

Use the Distributive Property to prove that these algebraic expressions are equivalent. Explain your answer in one sentence for each expression and show your work.

Expression 1: (x−5)(x+3)
(
𝑥
−
5
)
(
𝑥
+
3
)


Expression 2: 2x2−x2−2x−20+5

Answer 1

To prove that the expressions \( (x - 5)(x + 3) \) and \( 2x^2 - x^2 - 2x - 20 + 5 \) are equivalent using the Distributive Property, we will expand Expression 1 and simplify Expression 2.

### Step 1: Expand Expression 1
Using the Distributive Property (FOIL method):

\[
(x - 5)(x + 3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3
\]

Calculating each part, we have:

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

Combining like terms:

\[
= x^2 - 2x - 15
\]

### Step 2: Simplify Expression 2
Now we simplify Expression 2:

\[
2x^2 - x^2 - 2x - 20 + 5
\]

Calculating \((2x^2 - x^2)\):

\[
= (2 - 1)x^2 - 2x - 20 + 5
\]

This simplifies to:

\[
= x^2 - 2x - 20 + 5
\]

Combining the constant terms \((-20 + 5)\):

\[
= x^2 - 2x - 15
\]

### Conclusion
Both expressions simplify to \( x^2 - 2x - 15 \).

Thus, we can conclude:

**Expression 1** simplifies to \( x^2 - 2x - 15 \), and **Expression 2** also simplifies to \( x^2 - 2x - 15 \), proving that both expressions are equivalent.