elementary number theory - Give a proof by cases that shows that $n(n ^2 − 1)(n + 2)$ is a multiple of $4$, for all integers $n$

Give a proof by cases that shows that $n(n ^2 − 1)(n + 2)$ is a multiple of $4$, for all integers $n$

I have already done case 1 where n is even. I am doing case 2 where n is odd and I'm a bit confuse how to finish off the problem..Uploaded a picture of my work..
Is this work sufficient enough to prove that it is a multiple of 4? The 6k would not have any affect of the result in general?

I feel it is wrong because of the 6k. Right now my thoughts are if (statement) is a multiple of 4, then (statement) is a multiple of 2 as well. And work from there.

My work