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.

Question

Expression 1: (x−5)(x+3)


Expression 2: 2x2−x2−2x−20+5


(4 points) write it in one sentence only

Answer 1

To prove that the expressions \((x−5)(x+3)\) and \(2x^2−x^2−2x−20+5\) are equivalent, we will first simplify Expression 1 using the Distributive Property:

\[ (x−5)(x+3) = x(x+3) - 5(x+3) = x^2 + 3x - 5x - 15 = x^2 - 2x - 15. \]

Next, we will simplify Expression 2:

\[ 2x^2 - x^2 - 2x - 20 + 5 = (2x^2 - x^2) - 2x - 20 + 5 = x^2 - 2x - 15. \]

Since both expressions simplify to \(x^2 - 2x - 15\), we can conclude that they are equivalent.