To simplify the expression [tex]\(\frac{x+2}{x^2 + 5x + 6} \div \frac{3x + 1}{x^2 - 9}\)[/tex], follow these steps:
1. Factorize the Denominators:
- Factor [tex]\(x^2 + 5x + 6\)[/tex]:
[tex]\[
x^2 + 5x + 6 = (x + 2)(x + 3)
\][/tex]
- Factor [tex]\(x^2 - 9\)[/tex] (which is a difference of squares):
[tex]\[
x^2 - 9 = (x + 3)(x - 3)
\][/tex]
2. Rewrite the Original Expression Using Factored Forms:
[tex]\[
\frac{x+2}{(x + 2)(x + 3)} \div \frac{3x + 1}{(x + 3)(x - 3)}
\][/tex]
3. Simplify the Division By Multiplying by the Reciprocal:
[tex]\[
\frac{x+2}{(x + 2)(x + 3)} \times \frac{(x + 3)(x - 3)}{3x + 1}
\][/tex]
4. Cancel Common Factors:
- The term [tex]\(x + 2\)[/tex] cancels in the numerator and denominator on the left part.
- The term [tex]\(x + 3\)[/tex] cancels in the numerator and denominator across the two parts.
After canceling out the common factors, we are left with:
[tex]\[
\frac{1}{x + 3} \times \frac{(x - 3)}{(3x + 1)}
\][/tex]
5. Combine the Remaining Factors:
[tex]\[
\frac{(x - 3)}{3x + 1}
\][/tex]
So, the simplest form of the given expression is:
[tex]\(
\frac{x-3}{3 x+1}
\)[/tex]
Thus, the correct answer is:
B. [tex]\(\frac{x-3}{3 x+1}\)[/tex]
