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

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Expression 1: (x−5)(x+3)

Expression 2: 2x^2−x^2−2x−20+5

Answer 1

Expression 1: (x-5)(x+3) can be expanded using the distributive property to give x^2 + 3x - 5x - 15, which simplifies to x^2 - 2x - 15.

Expression 2: Using the commutative property for addition, we can rearrange the terms 2x^2, -x^2, -2x, -20, and 5 to give 2x^2 - x^2 - 2x - 20 + 5.

By combining like terms -x^2 and 2x^2, and combining the constants -20 and 5, we can simplify Expression 2 to x^2 - 2x - 15, which is equal to Expression 1.