Simplify \( [1 ÷ (x^2 + 3x + 2)] + [1 ÷ (x^2 + 5x + 6)] \)

A.

\( \frac{2}{(x + 1)^2} \)

B.

\( \frac{2}{(x + 1)(x + 2} \)

C.

\( \frac{2}{(x + 1)(x + 2} \)

D.

\( \frac{2}{(x + 1)(x + 3} \)

Correct answer is D

\( [1 ÷ (x^2 + 3x + 2)] + [1 ÷ (x^2 + 5x + 6)] \)

= \( 1 ÷ (x^2 + 3x + 2) + [1 ÷ (x^2 +5x + 6)]\)

= \( [1 ÷ ((x^2 + x) + (2x + 2) )] + [1 ÷ ((x^2 + 3x) + (2x + 6) )] \)

= [1 ÷ (x(x + 2) + 2(x +1))] + [1 ÷ (x(x + 3) +2(x + 3) )]

= [1 ÷ (x + 1)(x + 2)] + [1 ÷ ((x + 3) + (x + 2))]

=((x + 3) + (x + 1)) ÷ (x + 1)(x + 2)(x + 3)

Using the L.C.M

=((x + x + 3 + 1)) ÷ (x + 1)(x + 2)(x + 3)

=(2x+4)/(x+1)(x+2)(x+3) =2(x+2)/(x+1)(x+2)(x+3)

= \( \frac{2 }{(x + 1)(x + 3)} \)