Let's perform the operation and simplify the given expression step-by-step.
The given fractions are:
[tex]\[
\frac{x-3}{2x-8} \quad \text{and} \quad \frac{6x^2-96}{x^2-9}
\][/tex]
### Step 1: Factorize the Numerators and Denominators
1. Factorize [tex]\(6x^2 - 96\)[/tex]:
[tex]\[
6x^2 - 96 = 6(x^2 - 16) = 6(x - 4)(x + 4)
\][/tex]
2. Factorize [tex]\(x^2 - 9\)[/tex]:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
3. Factorize [tex]\(2x - 8\)[/tex]:
[tex]\[
2x - 8 = 2(x - 4)
\][/tex]
Now, rewrite the original expression with the factorizations applied:
[tex]\[
\frac{x - 3}{2(x - 4)} \cdot \frac{6(x - 4)(x + 4)}{(x - 3)(x + 3)}
\][/tex]
### Step 2: Cancel Common Factors
Identify and cancel the common factors from both the numerator and the denominator.
In the numerator and the denominator, we have:
[tex]\[
\frac{(x - 3) \cdot 6(x - 4)(x + 4)}{2(x - 4) \cdot (x - 3)(x + 3)}
\][/tex]
The common factors [tex]\((x - 3)\)[/tex] and [tex]\((x - 4)\)[/tex] can be canceled out:
[tex]\[
\frac{6(x + 4)}{2(x + 3)}
\][/tex]
### Step 3: Simplify the Remaining Expression
Further simplify the fraction:
[tex]\[
\frac{6(x + 4)}{2(x + 3)} = \frac{6}{2} \cdot \frac{(x + 4)}{(x + 3)} = 3 \cdot \frac{(x + 4)}{(x + 3)}
\][/tex]
Thus, the simplified result is:
[tex]\[
\frac{3(x + 4)}{(x + 3)}
\][/tex]
Hence, the detailed, step-by-step solution to the given problem is:
[tex]\[
\frac{3(x + 4)}{(x + 3)}
\][/tex]
