Use the Distributive Property to verify which expression is equivalent to

Question

$(x-5)(x-3)$ . (1 point)
$3x^{2}-4x^{2}-3x-15$
$x^{2}-2x+15$
$x^{2}-8x+15$
$2x-2$

Answer 1

To verify which expression is equivalent to \((x-5)(x-3)\), we will use the Distributive Property to expand the expression:

\[ (x-5)(x-3) = x(x-3) - 5(x-3) \]

Expanding each term:

1. \(x(x - 3) = x^2 - 3x\)
2. \(-5(x - 3) = -5x + 15\)

Now, combine these results:

\[ x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \]

Thus, the expansion gives us:

\[ (x-5)(x-3) = x^2 - 8x + 15 \]

Now, let’s compare \(x^2 - 8x + 15\) with the given options:

1. \(3x^{2}-4x^{2}-3x-15\) simplifies to \(-x^2 - 3x - 15\)
2. \(x^{2}-2x+15\) remains \(x^2 - 2x + 15\)
3. \(x^{2}-8x+15\) matches exactly \(x^2 - 8x + 15\)
4. \(2x-2\) remains \(2x - 2\)

The expression that is equivalent to \((x-5)(x-3)\) is:

\[ \boxed{x^{2}-8x+15} \]