To simplify the given expression:
[tex]\[
\frac{x+2}{x^2 + 5x + 6} \div \frac{3x+1}{x^2 - 9}
\][/tex]
Follow these steps:
1. Rewrite the division as multiplication by the reciprocal:
[tex]\[
\frac{x+2}{x^2 + 5x + 6} \times \frac{x^2 - 9}{3x+1}
\][/tex]
2. Factorize the denominators and numerators if possible:
- [tex]\( x^2 + 5x + 6 \)[/tex] can be factorized as [tex]\( (x + 2)(x + 3) \)[/tex].
- [tex]\( x^2 - 9 \)[/tex] is a difference of squares and can be factorized as [tex]\( (x + 3)(x - 3) \)[/tex].
Thus, the expression becomes:
[tex]\[
\frac{x+2}{(x + 2)(x + 3)} \times \frac{(x + 3)(x - 3)}{3x+1}
\][/tex]
3. Cancel out the common terms in the numerator and the denominator:
- The [tex]\( x + 2 \)[/tex] terms cancel out.
- The [tex]\( x + 3 \)[/tex] terms cancel out.
This simplifies to:
[tex]\[
\frac{1}{x + 3} \times \frac{x - 3}{3x + 1} = \frac{x - 3}{3x + 1}
\][/tex]
So, the simplest form of the given expression is:
[tex]\[
\frac{x - 3}{3x + 1}
\][/tex]
Hence, the correct answer is:
B. [tex]\(\frac{x-3}{3 x+1}\)[/tex]
