I claim that for all $n>4$ we have $n^2<2^n$.It can be showed by induction. The base case $n=5$ clearly holds true, so it suffices to show $\left(\frac{n+1}{n}\right)^2<2$. But for all $x>n$, we have $\frac{x+1}{x} < \frac{n+1}{n}$, so it suffices to show this holds true for $n=5$, and it does. Clearly $n^2=2^n$ cannot hold true for negative values of $n$