To verify which expression is equivalent to \((x-5)(x-3)\) using the Distributive Property, we will expand the original expression:
\[ (x-5)(x-3) = x(x-3) - 5(x-3) \]
Now, distribute \(x\) and \(-5\):
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Distributing \(x\): \[ x \cdot (x-3) = x^2 - 3x \]
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Distributing \(-5\): \[ -5 \cdot (x-3) = -5x + 15 \]
Now, combine the results:
\[ x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \]
Thus, the expression simplifies to:
\[ x^2 - 8x + 15 \]
Now, let's compare this result with the given options:
A. \(2x - 2\) (Not equivalent)
B. \(3x^2 - 4x^2 - 3x - 15\) (Incorrect since it simplifies to a different expression)
C. \(x^2 - 2x + 15\) (Not equivalent)
D. \(x^2 - 8x + 15\) (Equivalent)
The correct answer is:
D. \(x^2 - 8x + 15\)