Certainly! Let's solve this question step-by-step.
We need to determine the product of the following two expressions:
[tex]\[
\frac{x^2 - 16}{2x + 8} \cdot \frac{x^3 - 2x^2 + x}{x^2 + 3x - 4}
\][/tex]
### Step 1: Simplify Each Expression
First, let's simplify each fraction separately.
Simplifying the first fraction:
[tex]\[
\frac{x^2 - 16}{2x + 8}
\][/tex]
- Notice that \( x^2 - 16 \) is a difference of squares, which can be factored as \( (x + 4)(x - 4) \).
- Also, the denominator \( 2x + 8 \) can be factored out as \( 2(x + 4) \).
So, the first fraction can be rewritten as:
[tex]\[
\frac{(x + 4)(x - 4)}{2(x + 4)}
\][/tex]
- We can cancel out the \( x + 4 \) terms in the numerator and denominator:
[tex]\[
\frac{x - 4}{2}
\][/tex]
Simplifying the second fraction:
[tex]\[
\frac{x^3 - 2x^2 + x}{x^2 + 3x - 4}
\][/tex]
- The numerator \( x^3 - 2x^2 + x \) can be factored by taking out the common factor \( x \):
[tex]\[
x(x^2 - 2x + 1)
\][/tex]
- The quadratic \( x^2 - 2x + 1 \) is a perfect square, so it can be factored as:
[tex]\[
x(x - 1)^2
\][/tex]
Thus, the numerator becomes \( x(x - 1)^2 \).
- The denominator \( x^2 + 3x - 4 \) is the quadratic expression which can be factored into:
[tex]\[
(x + 4)(x - 1)
\][/tex]
So, the second fraction can be rewritten as:
[tex]\[
\frac{x(x - 1)^2}{(x + 4)(x - 1)}
\][/tex]
- We can cancel out an \( x - 1 \) term in the numerator and the denominator:
[tex]\[
\frac{x(x - 1)}{x + 4}
\][/tex]
### Step 2: Calculate the Product of the Simplified Expressions
Now, we multiply the simplified forms of the two fractions:
[tex]\[
\left(\frac{x - 4}{2}\right) \cdot \left(\frac{x(x - 1)}{x + 4}\right)
\][/tex]
We multiply the numerators together and the denominators together:
[tex]\[
=\frac{(x - 4) \cdot x \cdot (x - 1)}{2 \cdot (x + 4)}
\][/tex]
Simplifying the expression, we get:
[tex]\[
\frac{x(x - 4)(x - 1)}{2(x + 4)}
\][/tex]
This matches one of the given options exactly:
[tex]\[
\frac{x(x - 4)(x - 1)}{2(x + 4)}
\][/tex]
### Step 3: Identify the Correct Answer from the Choices
[tex]\[
\boxed{\frac{x(x - 4)(x - 1)}{2(x + 4)}}
\][/tex]
(-) \(\frac{x(x-1)}{2}\)
\(\frac{(x+4)(x-4)}{2 x(x-1)}\)
\(\frac{(x-4)(x-1)}{2 x(x+4)}\)
### Conclusion
By comparing the simplified product with the given choices, we find that the correct equation is:
[tex]\[
\boxed{\frac{x(x - 4)(x - 1)}{2(x + 4)}}
\][/tex]
Therefore, the product is:
[tex]\[
\frac{x(x - 4)(x - 1)}{2(x + 4)}
\][/tex]
The correct answer is the first choice:
[tex]\[
\boxed{\frac{x(x - 4)(x - 1)}{2(x + 4)}}
\][/tex]
