Direct Proof on Divisibilty Using Induction proof makes sense to me and know how to do, but I am having a problem in using a direct proof for practice problem that was given to us.
The problem is:
For all natural numbers $n$, $2n^3 + 6n^2 + 4n$ is divisible by 4.
We are to use direct proof as a way proving it. I have no clue where to start. 

For all $\,n\!:\,$  $\,4\mid 2f(n)$ $\!\iff\! 2\mid f(n)$ $\!\iff\! f(0)\equiv 0\equiv f(1)\pmod{\!2}$ $\!\iff 2\mid f(0),f(1)$ 
for any polynomial $\,f(x)\,$ with integer coefficients (see also the Parity Root Test).
This applies to $ $ OP $\ f(x) = n^3+3n^2+2n\,$ since $\ 2\mid f(0) = 0,\ $ and $\ 2\mid f(1) = 6$. 
