I would like some help with this question:

Q: x^2+x+1=0 (mod 13)

I can't factorise the LHS as it stands, so I changed the 0 to 13, rearrange to give:

x^2+x-12=0 (mod 13)

(x+4)(x-3)=0 (mod 13)

so:

x=-4 or 3 (mod 13)

Does my method work or have I gone wrong somewhere? Pointers would be greatly appreciated.