Answer: D
Step-by-step explanation:
The given expression is
(x + 2)/(x^2 + 5x + 6) ÷ (3x + 1)/(x^2 - 9)
We would simplify each term in the expression.
Considering (x^2 + 5x + 6), this is a quadratic expression. We would simplify by applying the method of factorization. The first step is to multiply x^2 with 6. It becomes 6x^2. We would find two terms such that their sum or difference is 5x and their product is 6x^2. The terms are 2x and 3x. By replacing 5x with 2x + 3x, we have
x^2 + 2x + 3x + 6
Factorize by grouping, it becomes
x(x + 2) + 3(x + 2)
(x + 2)(x + 3)
Also,
(x^2 - 9) can be written as (x + 3)(x - 3)
The original expression becomes
(x + 2)/(x + 2)(x + 3) ÷ (3x + 1)/(x + 3)(x - 3)
1/(x + 3) ÷ (3x + 1)/(x + 3)(x - 3)
If we flip (3x + 1)/(x + 3)(x - 3) such that it becomes (x + 3)(x - 3)/(3x + 1), the division sign becomes multiplication. Thus, the expression becomes
1/(x + 3) * (x + 3)(x - 3)/(3x + 1)
(x + 3) cancels out
The final expression is
D. (x - 3)/(3x + 1)