To verify which expression is equivalent to \((x-5)(x-3)\), we will first expand it using the distributive property (also known as the FOIL method for binomials).
\[ (x-5)(x-3) = x \cdot x + x \cdot (-3) - 5 \cdot x - 5 \cdot (-3) \]
Now let's calculate each part:
- \(x \cdot x = x^2\)
- \(x \cdot (-3) = -3x\)
- \(-5 \cdot x = -5x\)
- \(-5 \cdot (-3) = +15\)
Now we can combine these results:
\[ x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \]
So, the expression we derived from \((x - 5)(x - 3)\) is:
\[ x^2 - 8x + 15 \]
Now let's compare this with the options provided:
A. \(3x^2 - 4x^2 - 3x - 15\)
This can simplify to:
\[ (3x^2 - 4x^2) - 3x - 15 = -x^2 - 3x - 15 \quad \text{(not equivalent)} \]
B. \(r^2 - 2x + 15\)
This includes \(r\) instead of \(x\) and does not match \(x^2 - 8x + 15\) \quad \text{(not equivalent)}.
C. \(2x - 2\)
This is a linear expression and not quadratic \quad \text{(not equivalent)}.
D. \(r^2 - 8x + 15\)
Similarly, this includes \(r\) instead of \(x\) and also does not match \quad \text{(not equivalent)}.
None of the options A, B, C, or D are equivalent to \(x^2 - 8x + 15\).
The correct equivalent expression is not provided among the choices. The correct equivalent expression obtained from the multiplication is \(x^2 - 8x + 15\).