This identity can be easily proved by expanding the right side and then consolidating: \begin{align} &(a^2+2b^2-2ab)(a^2+2b^2+2ab)\\ &= (a^2+2b^2)^2-(2ab)^2\\ &= a^4 + 4b^4 - 4a^2b^2 - 4a^2b^2\\ &= a^4 + 4b^4 \end{align}