To find the quotient of the given rational expressions:
[tex]\[
\frac{x^2 - 36}{x + 8} \div \frac{x^2 + 12x + 36}{4x + 32}
\][/tex]
we need to follow these steps:
### Step 1: Rewrite the Division as Multiplication
Instead of dividing by the second rational expression, we multiply by its reciprocal:
[tex]\[
\frac{x^2 - 36}{x + 8} \times \frac{4x + 32}{x^2 + 12x + 36}
\][/tex]
### Step 2: Factorize the Expressions
Factor each polynomial in the expressions:
- For [tex]\(x^2 - 36\)[/tex]:
[tex]\[
x^2 - 36 = (x - 6)(x + 6)
\][/tex]
- For [tex]\(x^2 + 12x + 36\)[/tex]:
[tex]\[
x^2 + 12x + 36 = (x + 6)^2
\][/tex]
- For [tex]\(4x + 32\)[/tex]:
[tex]\[
4x + 32 = 4(x + 8)
\][/tex]
So the expression becomes:
[tex]\[
\frac{(x - 6)(x + 6)}{x + 8} \times \frac{4(x + 8)}{(x + 6)^2}
\][/tex]
### Step 3: Simplify the Expression
Now, we can simplify by canceling out the common terms in the numerator and the denominator:
- The [tex]\(x + 8\)[/tex] terms cancel out.
- One [tex]\(x + 6\)[/tex] term in the numerator and [tex]\(x + 6\)[/tex] terms in the denominator cancel out.
After cancellation:
[tex]\[
\frac{(x - 6)}{1} \times \frac{4}{x + 6} = \frac{4(x - 6)}{x + 6}
\][/tex]
### Step 4: Write Down the Simplified Quotient
The simplified quotient is:
[tex]\[
\frac{4(x - 6)}{x + 6}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\frac{4(x-6)}{x+6}}
\][/tex]
This corresponds to option A.
