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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.