$$ (a^4-2b^4)^4 = (a^4-2b^4)^2 \cdot (a^4-2b^4)^2 $$

Find $ \left(a^4-2b^4\right)^2 $ using formula.

$$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$

where $ A = \color{blue}{ a^4 } $ and $ B = \color{red}{ 2b^4 }$.

$$ \begin{aligned}\left(a^4-2b^4\right)^2 = \color{blue}{\left( a^4 \right)^2} -2 \cdot a^4 \cdot 2b^4 + \color{red}{\left( 2b^4 \right)^2} = a^8-4a^4b^4+4b^8\end{aligned} $$

Multiply each term of $ \left( \color{blue}{a^8-4a^4b^4+4b^8}\right) $ by each term in $ \left( a^8-4a^4b^4+4b^8\right) $.

$$ \left( \color{blue}{a^8-4a^4b^4+4b^8}\right) \cdot \left( a^8-4a^4b^4+4b^8\right) = \\ = a^{16}-4a^{12}b^4+4a^8b^8-4a^{12}b^4+16a^8b^8-16a^4b^{12}+4a^8b^8-16a^4b^{12}+16b^{16} $$

Combine like terms:

$$ a^{16} \color{blue}{-4a^{12}b^4} + \color{red}{4a^8b^8} \color{blue}{-4a^{12}b^4} + \color{green}{16a^8b^8} \color{orange}{-16a^4b^{12}} + \color{green}{4a^8b^8} \color{orange}{-16a^4b^{12}} +16b^{16} = \\ = a^{16} \color{blue}{-8a^{12}b^4} + \color{green}{24a^8b^8} \color{orange}{-32a^4b^{12}} +16b^{16} $$$$ \left( 8a^2b^6 \right)^2 = 8^2 \left( a^2 \right)^2 \left( b^6 \right)^2 = 64a^4b^{12} $$

Combine like terms:

$$ a^{16}-8a^{12}b^4+24a^8b^8 \color{blue}{-32a^4b^{12}} +16b^{16}+ \color{blue}{64a^4b^{12}} = a^{16}-8a^{12}b^4+24a^8b^8+ \color{blue}{32a^4b^{12}} +16b^{16} $$