Solving the congruence $3x^2 + 6x + 1 \equiv 0 \pmod {19}$

Question

I tried to solve this equation but without a success:

$3x^{2}+6x+1 \equiv 0 \pmod {19}$

I concluded hat $x(x+2)\equiv 6 \pmod{19}$, the only way i think to solve this is by just trying all the options. But there must be a more efficiant way.

I would like to get help with that, thanks

Answer 1

Complete the square: modulo $19$, we have $$(x+1)^2-1\equiv 6\iff (x+1)^2\equiv 7\iff x+1\equiv \pm 8.$$