To prove that the expressions \((x−5)(x+3)\) and \(2x^2−x^2−2x−20+5\) are equivalent using the Distributive Property, we will expand Expression 1 and simplify Expression 2.
Step 1: Expand Expression 1
\[ (x−5)(x+3) = x(x+3) - 5(x+3) \]
Distributing \(x\) and \(-5\):
\[ = x^2 + 3x - 5x - 15 \]
Combine like terms:
\[ = x^2 - 2x - 15 \]
Step 2: Simplify Expression 2
Expression 2 is given as:
\[ 2x^2−x^2−2x−20+5 \]
Combine like terms:
\[ = (2x^2 - x^2) - 2x - 20 + 5 \]
This simplifies to:
\[ = x^2 - 2x - 15 \]
Conclusion:
Now we can see that both expressions equal \(x^2 - 2x - 15\). Thus, we have shown that the expressions \((x−5)(x+3)\) and \(2x^2−x^2−2x−20+5\) are equivalent because they simplify to the same polynomial \(x^2 - 2x - 15\).
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