<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] You don't need induction to prove this.
$f(n)=n^4+2n^3+n^2 = n^2\cdot(n^2+2n+1) = n^2\cdot(n+1)^2$
Case 1: If $n$ is even, $n=2k, n^2=2k\cdot2k=4k^2$, now $4k^2\cdot(n+1)^2$, which is obvious that is divisible by 4
Case 2: If $n$ is odd then $n+1$ is even, let $m=n+1$, $m=2k, \ m^2=2k\cdot2k = 4k^2$, if we change in our first formula $n^2\cdot4k^2$, which is obvious that is divisible by 4.
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