Let's use the distributive property to expand the expression \((x-5)(x-3)\) and then check which of the given expressions is equivalent to it.
First, let's expand \((x-5)(x-3)\) using the distributive property (also known as FOIL for binomials):
\[ (x-5)(x-3) = x \cdot x + x \cdot (-3) + (-5) \cdot x + (-5) \cdot (-3) \]
Calculating each term:
- \(x \cdot x = x^2\)
- \(x \cdot (-3) = -3x\)
- \((-5) \cdot x = -5x\)
- \((-5) \cdot (-3) = +15\)
Now, add them together:
\[ x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \]
Now, we've determined that:
\[ (x-5)(x-3) = x^2 - 8x + 15 \]
Next, let's check which of the given expressions are equivalent to \(x^2 - 8x + 15\):
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For \(x^2 - 2x + 15\):
- This is not equivalent because the middle term \(-2x\) is different.
-
For \(3x^2 - 4x^2 - 3x - 15\):
- Simplifying \(3x^2 - 4x^2 - 3x - 15\) gives \(-x^2 - 3x - 15\).
- This is also not equivalent to \(x^2 - 8x + 15\).
-
For \(2x - 2\):
- This is not equivalent either, as it is simply a linear expression, while we have a quadratic with mixed terms.
-
For \(x^2 - 8x + 15\):
- This matches exactly with our result from expanding \((x-5)(x-3)\).
Thus, from the given options, the equivalent expression to \((x-5)(x-3)\) is:
\[ x^2 - 8x + 15 \]