To prove that the two expressions are equivalent using the Distributive Property, we will simplify both expressions individually.
### Expression 1: \((x - 5)(x + 3)\)
Using the Distributive Property (also known as the FOIL method for binomials), we will expand the expression:
1. **First**: \(x \cdot x = x^2\)
2. **Outer**: \(x \cdot 3 = 3x\)
3. **Inner**: \(-5 \cdot x = -5x\)
4. **Last**: \(-5 \cdot 3 = -15\)
Now, we combine these results:
\[
x^2 + 3x - 5x - 15 = x^2 - 2x - 15
\]
### Expression 2: \(2x^2 - x^2 - 2x - 20 + 5\)
Next, we simplify this expression:
1. Combine like terms:
- The \(2x^2\) and \(-x^2\) combine to give \(x^2\).
- Combine \(5\) and \(-20\): \(5 - 20 = -15\).
So we rewrite it as:
\[
x^2 - 2x - 15
\]
### Final Comparison
Now we have:
- Expression 1: \(x^2 - 2x - 15\)
- Expression 2: \(x^2 - 2x - 15\)
Since both expressions simplify to \(x^2 - 2x - 15\), we can conclude that they are equivalent.
### Explanation
Both expressions expand or simplify to \(x^2 - 2x - 15\), thus confirming their equivalence through the application of the Distributive Property.