Select the correct answer.

Assuming that no denominator equals zero, what is the simplest form of this expression?

[tex]\[
\frac{x+2}{x^2+5x+6} \div \frac{3x+1}{x^2-9}
\][/tex]

A. [tex]\(\frac{1}{(x+3)(x-3)}\)[/tex]

B. [tex]\(\frac{x-3}{3x+1}\)[/tex]

C. [tex]\(\frac{3x+1}{x-3}\)[/tex]

D. [tex]\(\frac{1}{3x+1}\)[/tex]



Answer :

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]