If $x\equiv2\pmod{3}$ prove that $3|4x^2+2x+1$ I've tried many different things to get a factor of $k-2$ but keep failing.
If $x\equiv2\pmod{3}$ prove that $3 \mid 4x^2+2x+1$

So let's track each term in $4x^2 + 2x + 1$. 
Since $x \equiv 2 \pmod{3}$, we get the following:
$$ 4x^2 \equiv 16\equiv 1 \pmod 3$$
$$2x \equiv 4 \equiv 1 \pmod 3$$
$$1 \equiv 1 \pmod 3$$
Adding these together, we get:
$$4x^2 + 2x + 1 \equiv 0 \pmod 3$$
We conclude that $3\mid (4x^2+2x+1)$.
