<s>[INST] Prove by induction that $n^4+2n^3+n^2$ is divisible by 4 
I'm trying to prove by induction that $n^4+2n^3+n^2$ is divisible by 4.

I know that P(1) it's true. Then $ n=k, P(k):k^4+2k^3+k^2=4w$ it's true by the hypothesis of induction. 
When I tried to prove $n=k+1, P(k+1):(k+1)^4+(k+1)^3+(k+1)^2 = 4t$,
$$k^4+4k^3+6k^2+4k+1+2k^3+6k^2+6k+2+k^2+2k+1 = 4t.$$
$$(k^4+2k^3+k^2)+4k^3+6k^2+4k+1+6k^2+6k+2+2k+1 = 4t.$$

Can I replace $(k^4+2k^3+k^2)$ of the previous expression by $4w$ and then continue the proof? Is this correct?
  Thanks.

 [/INST] As an alternative to proving via induction, consider the following hints:
Hint #1: $n^4 + 2n^3 + n^2 = n^2(n^2 + 2n + 1) = \bigg(n(n+1)\bigg)^2$
Hint #2: The parity of $n$ is different from that of $n+1$.
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