Answer
$x=-\dfrac{7}{5}$
Work Step by Step
The factored form of the given equation, $
\dfrac{x+2}{x^2+7x+10}=\dfrac{1}{3x+6}-\dfrac{1}{x+5}
,$ is
\begin{array}{l}\require{cancel}
\dfrac{x+2}{(x+5)(x+2)}=\dfrac{1}{3(x+2)}-\dfrac{1}{x+5}
.\end{array}
Multiplying both sides by the $LCD=
3(x+5)(x+2)
,$ then the solution to the given equation is
\begin{array}{l}\require{cancel}
3(x+2)=(x+5)(1)-3(x+2)(1)
\\\\
3x+6=x+5-3x-6
\\\\
3x-x+3x=5-6-6
\\\\
5x=-7
\\\\
x=-\dfrac{7}{5}
.\end{array}
