\[x^{8} + 64 =\]
\[= \left( x^{4} \right)^{2} + 16x^{4} + 64 - 16x^{4} =\]
\[= \left( x^{4} + 8 \right)^{2} - 16x^{4} =\]
\[= \left( x^{4} + 8 - 4x^{2} \right)\left( x^{4} + 8 + 4x^{2} \right)\]