Answer
See below.
Work Step by Step
Let $n$ be any integer, we have:
case1: it is even, $n=2k$, thus $n^4=16k^4=8(2k^4)$, and $m=2k^4$;
case2: it is odd, $n=2k+1$, thus
$n^4=(2k+1)^2(2k+1)^2\\
=(4k^2+4k+1)(4k^2+4k+1)\\
=16k^4+32k^3+24k^2+8k+1\\
=8(2k^4+4k^3+3k^2+k)+1$,
and $m=2k^4+4k^3+3k^2+k$;
thus, in any case, we can write $n^4$ in the form of $8m$ or $8m+1$.
