Answer
See below.
Work Step by Step
Let $n$ be any integer, we have:
case1: it is even, $n=2k$, and $n^2+5=4k^2+5=4(k^2+1)+1$, thus it is not divisible by 4;
case2: it is odd, $n=2k+1$, and $n^2+5=4k^2+4k+1+5=4(k^2+k+1)+2$, thus it is not divisible by 4;
thus, in any case, $n^2+5$ is not divisible by 4.