Question

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. Expression 1: (x−5)(x+3) Expression 2: 2x2−x2−2x−20+5

Answer 1

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.