<s>[INST] 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 $3 \mid 4x^2+2x+1$
 [/INST] Another way to prove it:
Since $x \equiv 2 \mod 3$, $\exists k \in \mathbb{Z}$ such that $x=2+3k$. Therefore:
$$4x^2+2x+1=4(2+3k)^2+2(2+3k)+1=21+54 k+36 k^2 = 3(12k^2+18k+7)$$
So clearly $3 \mid (4x^2+2x+1)$.
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