in 1-3 sentences explain: 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: 2x^2−x^2−2x−20+5

Answer 1

To prove that the expressions \((x−5)(x+3)\) and \(2x^2−x^2−2x−20+5\) are equivalent, we can simplify both expressions.

For Expression 1: Using the Distributive Property, we have: \[ (x−5)(x+3) = x^2 + 3x - 5x - 15 = x^2 - 2x - 15. \]

For Expression 2: Now simplify it: \[ 2x^2 - x^2 - 2x - 20 + 5 = (2x^2 - x^2) - 2x - 20 + 5 = x^2 - 2x - 15. \]

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