Let’s solve the given problem step-by-step.
We need to find the quotient of the following rational expressions:
[tex]\[
\frac{x^2 + 5x + 6}{x - 6} \div \frac{x^2 - 9}{2x - 12}
\][/tex]
To divide by a fraction, we multiply by its reciprocal:
[tex]\[
\frac{x^2 + 5x + 6}{x - 6} \times \frac{2x - 12}{x^2 - 9}
\][/tex]
Next, let's factorize all the polynomials involved:
1. The numerator [tex]\( x^2 + 5x + 6 \)[/tex]:
[tex]\[
x^2 + 5x + 6 = (x + 2)(x + 3)
\][/tex]
2. The denominator [tex]\( x - 6 \)[/tex] does not need to be factorized.
3. The numerator [tex]\( 2x - 12 \)[/tex]:
[tex]\[
2x - 12 = 2(x - 6)
\][/tex]
4. The denominator [tex]\( x^2 - 9 \)[/tex]:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
Substituting these factorizations into the expression, we get:
[tex]\[
\frac{(x + 2)(x + 3)}{x - 6} \times \frac{2(x - 6)}{(x - 3)(x + 3)}
\][/tex]
Now, cancel out the common factors in the numerator and the denominator:
1. [tex]\( x - 6 \)[/tex] cancels out.
2. [tex]\( x + 3 \)[/tex] cancels out.
This leaves us with:
[tex]\[
\frac{(x + 2) \cdot 2}{x - 3}
\][/tex]
Simplifying further, we get:
[tex]\[
\frac{2(x + 2)}{x - 3}
\][/tex]
Thus, the quotient in reduced form is:
[tex]\[
\frac{2(x + 2)}{x - 3}
\][/tex]
The correct answer is:
[tex]\[
\boxed{A}
\][/tex]
