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.


You are not doing it wrongly but it's easier, with $f(n)=n^4+2n^3+n^2$ , to observe that $f(n)=(n+1)^2n^2,$ so $$f(n+1)-f(n)=(n+2)^2(n+1)^2-(n+1)^2n^2=$$ $$=(n+1)^2 ((n+2)^2-n^2)=$$ $$=(n+1)^2(4n+4)=4(n+1)^3$$ which is a multiple of $4$.
