To prove that the expressions \((x - 5)(x + 3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent, we'll expand Expression 1 and simplify Expression 2.
**Expression 1: \((x - 5)(x + 3)\)**
Using the Distributive Property (FOIL method):
\[
(x - 5)(x + 3) = x\cdot x + x\cdot 3 - 5\cdot x - 5\cdot 3
\]
\[
= x^2 + 3x - 5x - 15
\]
\[
= x^2 - 2x - 15
\]
**Expression 2: \(2x^2 - x^2 - 2x - 20 + 5\)**
Now, simplify Expression 2 by combining like terms:
\[
2x^2 - x^2 = x^2
\]
\[
-2x
\]
\[
-20 + 5 = -15
\]
Thus, we have:
\[
\text{Expression 2} = x^2 - 2x - 15
\]
Since both expressions simplify to \(x^2 - 2x - 15\), we conclude that they are equivalent.