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.

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**

\[
(x−5)(x+3) = x(x+3) - 5(x+3)
\]

Distributing \(x\) and \(-5\):

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

Combine like terms:

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

**Step 2: Simplify Expression 2**

Expression 2 is given as:

\[
2x^2−x^2−2x−20+5
\]

Combine like terms:

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

This simplifies to:

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

**Conclusion:**

Now we can see that both expressions equal \(x^2 - 2x - 15\). Thus, we have shown that the expressions \((x−5)(x+3)\) and \(2x^2−x^2−2x−20+5\) are equivalent because they simplify to the same polynomial \(x^2 - 2x - 15\).

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