\[\frac{31}{4\sqrt{2} + 1} = \frac{31 \cdot \left( 4\sqrt{2} - 1 \right)}{\left( 4\sqrt{2} + 1 \right)\left( 4\sqrt{2} - 1 \right)} =\]
\[= \frac{31 \cdot \left( 4\sqrt{2} - 1 \right)}{32 - 1} = \frac{31 \cdot \left( 4\sqrt{2} - 1 \right)}{31} =\]
\[= 4\sqrt{2} - 1\]