Factor the denominator of the third fraction:
[tex]x^2+5x+6=x^2+3x+2x+6=x(x+3)+2(x+3)=(x+3)(x+2).[/tex]
Then
[tex]\dfrac{1}{x+2}+\dfrac{1}{x+3}+\dfrac{1}{x^2+5x+6}=\dfrac{1}{x+2}+\dfrac{1}{x+3}+\dfrac{1}{(x+2)(x+3)}.[/tex]
The common denominator is [tex](x+2)(x+3),[/tex] so
[tex]\dfrac{1}{x+2}+\dfrac{1}{x+3}+\dfrac{1}{(x+2)(x+3)}=\dfrac{1\cdot (x+3)+1\cdot (x+2)+1}{(x+2)(x+3)}=\dfrac{2x+6}{(x+2)(x+3)}=\\ \\\dfrac{2(x+3)}{(x+2)(x+3)}=\dfrac{2}{x+2}.[/tex]
Answer: correct choice is A.
