Structures of Expressions Unit Test

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

### Step 1: Expand Expression 1
\[
(x-5)(x+3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3
\]
\[
= x^2 + 3x - 5x - 15
\]
\[
= x^2 - 2x - 15
\]

### Step 2: Simplify Expression 2
Now, let's simplify Expression 2:
\[
2x^2 - x^2 - 2x - 20 + 5
\]
Combine like terms:
\[
= (2x^2 - x^2) - 2x - 20 + 5
\]
\[
= x^2 - 2x - 15
\]

### Conclusion
After performing the calculations, we find that both expressions simplify to \(x^2 - 2x - 15\). Thus, we can conclude that the two expressions are equivalent.

**Answer:** Expression 1 simplifies to \(x^2 - 2x - 15\) and Expression 2 simplifies to \(x^2 - 2x - 15\), proving they are equivalent; both expressions ultimately equal \(x^2 - 2x - 15\).