\[\left( b^{n + 2} - c^{n - 2} \right)\left( b^{n + 2} + c^{n - 2} \right) =\]
\[= \left( b^{n + 2} \right)^{2} - \left( c^{n - 2} \right)^{2} =\]
\[= b^{2 \cdot (n + 2)} - c^{2 \cdot (n - 2)} =\]
\[= b^{2n + 4} - c^{2n - 4}.\]