Answer
$a=-\dfrac{7}{5}$
Work Step by Step
The given expression simplifies to
\begin{array}{l}
\dfrac{1}{a+5}=\dfrac{1}{3a+6}-\dfrac{a+2}{a^2+7a+10}
\\\\
\dfrac{1}{a+5}=\dfrac{1}{3(a+2)}-\dfrac{a+2}{(a+2)(a+5)}
\\\\
\dfrac{1}{a+5}=\dfrac{1}{3(a+2)}-\dfrac{1}{a+5}
\\\\
\dfrac{1}{a+5}=\dfrac{1}{3(a+2)}-\dfrac{1}{a+5}
\\\\
\dfrac{1}{a+5}+\dfrac{1}{a+5}=\dfrac{1}{3(a+2)}
\\\\
\dfrac{2}{a+5}=\dfrac{1}{3(a+2)}
\\\\
2[3(a+2)]=1(a+5)
\\\\
6(a+2)=1(a+5)
\\\\
6a+12=a+5
\\\\
6a-a=5-12
\\\\
5a=-7
\\\\
a=-\dfrac{7}{5}
.\end{array}
