Answer
$\dfrac{4(a-3)}{-3(a+6)}$
Work Step by Step
The given expression, $
\dfrac{\dfrac{5}{a+2}-\dfrac{1}{a-2}}{\dfrac{3}{2+a}+\dfrac{6}{2-a}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{\dfrac{5}{a+2}-\dfrac{1}{a-2}}{\dfrac{3}{a+2}-\dfrac{6}{a-2}}
\\\\=
\dfrac{\dfrac{(a-2)(5)-(a+2)(1)}{(a+2)(a-2)}}{\dfrac{(a-2)(3)-(a+2)(6)}{(a+2)(a-2)}}
\\\\=
\dfrac{\dfrac{(a-2)(5)-(a+2)(1)}{\cancel{(a+2)(a-2)}}}{\dfrac{(a-2)(3)-(a+2)(6)}{\cancel{(a+2)(a-2)}}}
\\\\=
\dfrac{(a-2)(5)-(a+2)(1)}{(a-2)(3)-(a+2)(6)}
\\\\=
\dfrac{5a-10-a-2}{3a-6-6a-12}
\\\\=
\dfrac{4a-12}{-3a-18}
\\\\=
\dfrac{4(a-3)}{-3(a+6)}
.\end{array}
