Question

use the distributive property to show that these expressions are equivalent
(x - 5)(x + 3) and 2x^2 - x^2 - 2x - 20 + 5
explain your answer in one sentence for each expression and show the steps of your work for each expression

Answer 1

To show that the expressions \((x - 5)(x + 3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent, we will simplify both expressions using algebraic methods, specifically the distributive property.

### Expression 1: \((x - 5)(x + 3)\)

1. Apply the distributive property (also known as the FOIL method for binomials):
\[
(x - 5)(x + 3) = x(x + 3) - 5(x + 3)
\]
2. Distributing \(x\):
\[
= x^2 + 3x
\]
3. Distributing \(-5\):
\[
= -5x - 15
\]
4. Combine the results:
\[
= x^2 + 3x - 5x - 15
\]
5. Combine like terms:
\[
= x^2 - 2x - 15
\]

### Expression 2: \(2x^2 - x^2 - 2x - 20 + 5\)

1. Begin simplifying by combining like terms:
\[
= (2x^2 - x^2) - 2x - 20 + 5
\]
2. Combine \(2x^2\) and \(-x^2\):
\[
= x^2 - 2x - 20 + 5
\]
3. Combine constants \(-20 + 5\):
\[
= x^2 - 2x - 15
\]

### Conclusion:
The simplified form of both expressions, \((x - 5)(x + 3)\) and \(2x^2 - x^2 - 2x - 20 + 5\), results in \(x^2 - 2x - 15\), demonstrating that the two expressions are equivalent.

Thus, \((x - 5)(x + 3) = x^2 - 2x - 15\) and \(2x^2 - x^2 - 2x - 20 + 5 = x^2 - 2x - 15\) yield the same polynomial.