\[\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}} = \frac{\left( \sqrt{6} + \sqrt{2} \right)\left( \sqrt{6} + \sqrt{2} \right)}{\left( \sqrt{6} - \sqrt{2} \right)\left( \sqrt{6} + \sqrt{2} \right)} =\]
\[= \frac{\left( \sqrt{6} + \sqrt{2} \right)^{2}}{6 - 2} = \frac{6 + 2\sqrt{6 \cdot 2} + 2}{4} =\]
\[= \frac{8 + 2\sqrt{12}}{4} = \frac{8 + 4\sqrt{3}}{4} = 2 + \sqrt{3}.\]