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