To verify which expression is equivalent to \((x-5)(x-3)\) using the Distributive Property (also known as the FOIL method for binomials), we can expand the expression step by step.
Using the FOIL method:
- First: Multiply the first terms: \(x \cdot x = x^2\)
- Outside: Multiply the outside terms: \(x \cdot (-3) = -3x\)
- Inside: Multiply the inside terms: \(-5 \cdot x = -5x\)
- Last: Multiply the last terms: \(-5 \cdot (-3) = 15\)
Now combine all these results:
\[ x^2 - 3x - 5x + 15 \]
Combine the like terms:
\[ x^2 - 8x + 15 \]
So, the expression \((x-5)(x-3)\) simplifies to \(x^2 - 8x + 15\).
Now let's compare this with the provided responses:
- \(x^2 - 2x + 15\) (not equivalent)
- \(x^2 - 8x + 15\) (this matches)
- \(2x - 2\) (not equivalent)
- \(3x^2 - 4x^2 - 3x - 15\) (this does not simplify to the correct form)
Thus, the correct answer is:
\(x^2 - 8x + 15\).