I see that other answers apply the congruence method directly, but I think that unveiling this notation and seeing how it works are of some use still.
If $4\mid n$, say $n=4k$, then $$n^2+5=16k^2+25=4(4k^2+6)+1$$ is certainly not divisible by $4$.
If $n=4a+b$, where $b=1, 2, 3$. Then $$n^2+5=16a^2+8ab+b^2+5=4(4a^2+2ab)+b^2+5.$$ So if $n^2+5$ is divisible by $4$, then so is $b^2+1$. But this is impossible for $b=1, 2, 3$.
So the statement follows. In fact, this is nothing but untailing the details of the congruence notation, thus permitting one to see the power of this seemingly innocent congruence notation.
Barring mistakes, and thanks for the attention.