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: Expanding Expression 1

We will use the Distributive Property (also known as the FOIL method for binomials) to expand \((x - 5)(x + 3)\):

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

### Step 2: Simplifying Expression 2

Now let's simplify Expression 2: \(2x^2 - x^2 - 2x - 20 + 5\):

\[
2x^2 - x^2 - 2x - 20 + 5 = (2x^2 - x^2) - 2x + (-20 + 5)
\]
\[
= x^2 - 2x - 15
\]

### Conclusion

Both expressions simplify to \(x^2 - 2x - 15\), proving that \((x - 5)(x + 3) = 2x^2 - x^2 - 2x - 20 + 5\). Thus, we conclude that the two expressions are equivalent since both yield the same polynomial after simplification.