To verify which expression is equivalent to \((x - 5)(x - 3)\) using the Distributive Property (also known as the FOIL method for binomials), we need to expand this expression.
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Apply the Distributive Property (FOIL):
- First: \(x \cdot x = x^2\)
- Outside: \(x \cdot (-3) = -3x\)
- Inside: \(-5 \cdot x = -5x\)
- Last: \(-5 \cdot (-3) = 15\)
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Combine the results: \[ x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \]
So, \((x - 5)(x - 3) = x^2 - 8x + 15\).
Now, let's compare this with the answer choices:
A) \(2x - 2\) → Not equivalent.
B) \(x^2 - 8x + 15\) → This is the correct equivalent expression.
C) \(x^2 - 2x + 15\) → Not equivalent.
D) \(3x^2 - 4x^2 - 3x - 15\) → This simplifies to \(-x^2 - 3x - 15\), which is not equivalent.
Therefore, the expression that is equivalent to \((x - 5)(x - 3)\) is B) \(x^2 - 8x + 15\).