Let's simplify the given expression step by step:
The given expression to simplify is:
[tex]\[
\frac{8}{2x + 8} \cdot \frac{x^2 - 16}{4}
\][/tex]
### Step 1: Factorize the fractions
1. Factorize the denominator of the first fraction:
[tex]\[
2x + 8 = 2(x + 4)
\][/tex]
So, the first fraction becomes:
[tex]\[
\frac{8}{2(x + 4)}
\][/tex]
2. Notice the numerator of the second fraction is a difference of squares:
[tex]\[
x^2 - 16 = (x - 4)(x + 4)
\][/tex]
So, the second fraction becomes:
[tex]\[
\frac{(x - 4)(x + 4)}{4}
\][/tex]
### Step 2: Write the product of the fractions
Now we write the product of the fractions:
[tex]\[
\left( \frac{8}{2(x + 4)} \right) \cdot \left( \frac{(x - 4)(x + 4)}{4} \right)
\][/tex]
### Step 3: Multiply the numerators and the denominators
1. Multiply the numerators:
[tex]\[
8 \cdot (x - 4)(x + 4)
\][/tex]
2. Multiply the denominators:
[tex]\[
2(x + 4) \cdot 4 = 8(x + 4)
\][/tex]
So, the expression now becomes:
[tex]\[
\frac{8 \cdot (x - 4)(x + 4)}{8(x + 4)}
\][/tex]
### Step 4: Simplify the expression
Notice that the factor [tex]\(8\)[/tex] and [tex]\((x + 4)\)[/tex] in the numerator and the denominator can be cancelled out:
[tex]\[
\frac{8 \cdot (x - 4)(x + 4)}{8(x + 4)} = \frac{(x - 4)(x + 4)}{x + 4}
\][/tex]
[tex]\[
= x - 4
\][/tex]
### Step 5: Choose the correct answer
The simplified form of the given expression is:
[tex]\[
x - 4
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\text{C. } (x-4)}
\][/tex]
