<s>[INST] proof with induction $2^n + 4 > n^2 + 2n$ prove w/ full induction : $2^n + 4 > n^2 + 2n$.
Let's skip the other steps for a moment, I'm just getting stuck with the proof
the claim is: $2^{n+1} + 4 > (n+1)^2 + 2(n+1)$
and now the proof  $$2^{n+1} + 4 > 2 * 2^n + 4$$ $$> 2 * (n^2 + 2n)$$ $$> 2n^2 + 4n$$
i am stuck here
 [/INST] The claim is $2^{n+1} + 4 > (n+1)^2 + 2(n+1)$
Assuming it is true for $n$, that is $2^n + 4 > n^2 + 2n\to 2^n>n^2+2n-4$ we have
$$2^{n+1} + 4=2\cdot 2^n+4>2\left(n^2+2n-4\right)+4=2n^2+4n-4=(n^2+2n+1)+(2n+2)+n^2-6$$
$$>(n+1)^2+2(n+1);\;n>2$$
Proved
</s>