Answer
$\dfrac{-x^2+23x+38}{(x-2)(x+2)(x+2)}$
Work Step by Step
Using the $LCD=(x-2)(x+2)(x+2)$, the expression $
\dfrac{5}{x-2}+\dfrac{3}{x^2+4x+4}-\dfrac{6}{x+2}
$ simplifies to
\begin{array}{l}
\dfrac{5}{x-2}+\dfrac{3}{(x+2)(x+2)}-\dfrac{6}{x+2}
\\\\=
\dfrac{5(x+2)(x+2)+3(x-2)-6(x-2)(x+2)}{(x-2)(x+2)(x+2)}
\\\\=
\dfrac{5(x^2+4x+4)+3x-6-6(x^2-4)}{(x-2)(x+2)(x+2)}
\\\\=
\dfrac{5x^2+20x+20+3x-6-6x^2+24}{(x-2)(x+2)(x+2)}
\\\\=
\dfrac{(5x^2-6x^2)+(20x+3x)+(20-6+24)}{(x-2)(x+2)(x+2)}
\\\\=
\dfrac{-x^2+23x+38}{(x-2)(x+2)(x+2)}
.\end{array}
