when it comes to checking if a function is even or odd, it boils down to changing the argument, namely x = -x, and if the resulting function is the same as the original, then is even, if the resulting function is the same as the original but negative, is odd, if neither, well then neither :).
anyway, that said, let's first expand it and then plug in -x,
[tex] \bf f(x)=(x^2-8)^2\implies f(x)=(x^2-8)(x^2-8)\implies f(x)=\stackrel{FOIL}{x^4-16x^2+64}\\\\[-0.35em]~\dotfill\\\\f(-x)=(-x)^4-16(-x)^2+64\qquad \begin{cases}(-x)(-x)(-x)(-x)=x^4\\(-x)(-x)=x^2\end{cases}\\\\\\f(-x)=x^4-16x^2+64\impliedby \stackrel{\textit{same as the original}}{Even} [/tex]
