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 $n$? 


$$P \equiv n^4-2n^3-n^2+2n \equiv (n-2)\cdot(n-1)\cdot n\cdot(n+1)\equiv 0\mod 24 $$
The last one is because four consecutive numbers must include one number that is divisible by $4$, another number that is divisible by $2$, and one number that is divisible by $3$.
