\[\left( \sqrt{5} + 2\sqrt{2} \right)\left( \sqrt{5} - 2\sqrt{2} \right) =\]
\[= \left( \sqrt{5} \right)^{2} - \left( 2\sqrt{2} \right)^{2} = 5 - 4 \cdot 2 =\]
\[= 5 - 8 = - 3\]