To determine the product of the given fractions
[tex]\[
\frac{x^2 - 16}{2x + 8} \cdot \frac{x^3 - 2x^2 + x}{x^2 + 3x - 4},
\][/tex]
we need to simplify and multiply them. We will simplify each fraction step-by-step and then find the product.
Step 1: Simplify the first fraction:
[tex]\[
\frac{x^2 - 16}{2x + 8}
\][/tex]
Notice that [tex]\(x^2 - 16\)[/tex] is a difference of squares and can be factored as:
[tex]\[
x^2 - 16 = (x - 4)(x + 4)
\][/tex]
So the first fraction becomes:
[tex]\[
\frac{(x - 4)(x + 4)}{2(x + 4)}
\][/tex]
Cancel the common factor [tex]\((x + 4)\)[/tex]:
[tex]\[
\frac{(x - 4)}{2}
\][/tex]
Step 2: Simplify the second fraction:
[tex]\[
\frac{x^3 - 2x^2 + x}{x^2 + 3x - 4}
\][/tex]
Factor the numerator [tex]\(x^3 - 2x^2 + x\)[/tex]:
[tex]\[
x^3 - 2x^2 + x = x(x^2 - 2x + 1) = x(x - 1)^2
\][/tex]
Factor the denominator [tex]\(x^2 + 3x - 4\)[/tex]:
[tex]\[
x^2 + 3x - 4 = (x + 4)(x - 1)
\][/tex]
So the second fraction becomes:
[tex]\[
\frac{x(x - 1)^2}{(x + 4)(x - 1)}
\][/tex]
Cancel the common factor [tex]\((x - 1)\)[/tex]:
[tex]\[
\frac{x(x - 1)}{x + 4}
\][/tex]
Step 3: Multiply the simplified fractions from Step 1 and Step 2:
[tex]\[
\frac{x - 4}{2} \cdot \frac{x(x - 1)}{x + 4}
\][/tex]
Multiply the numerators and the denominators:
[tex]\[
\frac{(x - 4) \cdot x(x - 1)}{2(x + 4)}
\][/tex]
Simplify the product:
[tex]\[
\frac{x(x - 4)(x - 1)}{2(x + 4)}
\][/tex]
So the simplified form of the product is:
[tex]\[
\frac{x(x - 4)(x - 1)}{2(x + 4)}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\frac{x(x - 4)(x - 1)}{2(x + 4)}
\][/tex]
