<s>[INST] proof that $n^4+22n^3+71n^2+218n+384$ is divisible by $24$ Let $n$ be a positive integer number.

How do we show that the irreducible polynomial $n^4+22n^3+71n^2+218n+384$ is divisible by $24$ for all $n$? 

 [/INST] Because $$n^4+22n^3+71n^2+218n+384=n(n+1)(n+2)(n+3)+16n^3+60n^2+212n+384=$$
$$=n(n+1)(n+2)(n+3)+384+16(n^3-n)+60n^2+228n=$$
$$=n(n+1)(n+2)(n+3)+384+16(n-1)n(n+1)+60n(n+1)+168n.$$
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