$$\frac{x^{2}-4}{x^{2}+4x+4}\div\frac{x^{3}-8}{2x+4}$$
$\frac{2}{x^{2}+2x+4}$
$$\frac{\left(x^{2}-4\right)\left(2x+4\right)}{\left(x^{2}+4x+4\right)\left(x^{3}-8\right)}$$
$$\frac{2\left(x-2\right)\left(x+2\right)^{2}}{\left(x-2\right)\left(x+2\right)^{2}\left(x^{2}+2x+4\right)}$$
$$\frac{2}{x^{2}+2x+4}$$
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