\[\frac{x^{2} - 64}{4 + \sqrt{x + 8}} =\]
\[= \frac{(x - 8)(x + 8)\left( 4 - \sqrt{x + 8} \right)}{\left( 4 + \sqrt{x + 8} \right)\left( 4 - \sqrt{x + 8} \right)} =\]
\[= \frac{(x - 8)(x + 8)\left( 4 - \sqrt{x + 8} \right)}{16 - x - 8} =\]
\[= - (x + 8)\left( 4 - \sqrt{x + 8} \right) =\]
\[= (x + 8)(\sqrt{x + 8} - 4)\]