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 $3 \mid 4x^2+2x+1$

Hint $\ {\rm mod}\ 3\!:\ x\equiv 2\equiv -1\,\Rightarrow\, 1+2x+4x^2\equiv 1-2+4\equiv 3\equiv 0.\,$ 
Generally for polynomial $\ p(x),\,\ \ p(-1) = p_0-p_1+p_2-p_3+\cdots = $ alternating coef sum.
Alternatively $\,\color{#c00}{4\equiv1}\,\Rightarrow\, \color{#c00}{4x^2}\!+2x+1\equiv \color{#c00}{x^2}\!+2x+1\equiv (x\!+\!1)^2\,$ which has root $\,x\equiv-1\equiv 2.$
There are also many other ways to proceed, and all provide good exercise in modular arithmetic.
