To verify which expression is equivalent to \((x - 5)(x - 3)\) using the Distributive Property, we will expand the expression:
\[ (x - 5)(x - 3) = x \cdot x + x \cdot (-3) + (-5) \cdot x + (-5) \cdot (-3) \]
Now let's perform the multiplication:
- \(x \cdot x = x^2\)
- \(x \cdot (-3) = -3x\)
- \((-5) \cdot x = -5x\)
- \((-5) \cdot (-3) = 15\)
Now, we can combine all the terms:
\[ x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \]
So, \((x - 5)(x - 3)\) simplifies to:
\[ x^2 - 8x + 15 \]
Now, we can compare this result to the given options:
A. \(3x^2 - 4x^2 - 3x - 15\) = \( -x^2 - 3x - 15\) (not equivalent) B. \(2x - 2\) (not equivalent) C. \(x^2 - 2x + 15\) (not equivalent) D. \(x^2 - 8x + 15\) (equivalent)
Thus, the correct answer is:
D. \(x^2 - 8x + 15\)