Answer
$\dfrac{(x+2)(x+2)}{6(x-2)}$
Work Step by Step
Factoring the expressions and then cancelling the common factor/s between the numerator and the denominator, then the given expression, $
\dfrac{x^2-4}{3x+6}\div\dfrac{2x^2-8x+8}{x^2+4x+4}
$, simplifies to
\begin{array}{l}\require{cancel}
\dfrac{x^2-4}{3x+6}\cdot\dfrac{x^2+4x+4}{2x^2-8x+8}
\\\\=
\dfrac{(x+2)(x-2)}{3(x+2)}\cdot\dfrac{(x+2)(x+2)}{2(x^2-4x+4)}
\\\\=
\dfrac{(x+2)(x-2)}{3(x+2)}\cdot\dfrac{(x+2)(x+2)}{2(x-2)(x-2)}
\\\\=
\dfrac{(\cancel{x+2})(\cancel{x-2})}{3(\cancel{x+2})}\cdot\dfrac{(x+2)(x+2)}{2(\cancel{x-2})(x-2)}
\\\\=
\dfrac{(x+2)(x+2)}{6(x-2)}
.\end{array}
