Use the Distributive Property to prove that these algebraic expressions are equivalent. Explain your answer in 3 sentences for each expression and show your work. Please tell me how you got your answer and justify how you got your answer.

EXPRESSION 1: (x – 5) (x + 3)

EXPRESSION 2: 2x² - x² - 2x – 20 + 5

### Expression 1: \((x - 5)(x + 3)\)
We will apply the Distributive Property (also known as the FOIL method for binomials) to expand the expression.

\[
(x - 5)(x + 3) = x(x + 3) - 5(x + 3)
\]

Now distribute \(x\) and \(-5\) to both terms inside the parentheses:

\[
= x^2 + 3x - 5x - 15
\]

Combine like terms:

\[
= x^2 - 2x - 15
\]

Thus, \((x - 5)(x + 3) = x^2 - 2x - 15\).

### Expression 2: \(2x^2 - x^2 - 2x - 20 + 5\)
Simplify this expression by combining like terms.

\[
2x^2 - x^2 = x^2
\]

Next, combine constants:

\[
-20 + 5 = -15
\]

Thus, the simplified expression is:

\[
x^2 - 2x - 15
\]

### Conclusion:
Both expressions simplify to the same result: \(x^2 - 2x - 15\). Therefore, the expressions \((x - 5)(x + 3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent, as they simplify to the same polynomial.

Would you like more details or have any questions?

Here are 5 related questions to deepen understanding:
1. How does the Distributive Property apply to multiplying binomials?
2. What is the importance of combining like terms in simplifying expressions?
3. Can you factor \(x^2 - 2x - 15\) back into two binomials?
4. How does the FOIL method work in expanding binomials?
5. Why is it important to combine constants separately from variable terms?

**Tip:** Always simplify expressions step by step to avoid mistakes when combining terms!

Learn how to use the Distributive Property to prove that the algebraic expressions (x - 5)(x + 3) and 2x² - x² - 2x - 20 + 5 are equivalent. This detailed solution explains the key steps in expanding and simplifying polynomials. Suitable for students in Grades 8-10.

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