question on distinct solutions to congruences
I had 5 problems to find all distinct solutions or show there are none. I did all 5 using a table of x values and just testing each, basically brute forcing them. I seem to be all over the place trying to figure them out in a more efficient manner. Any suggestions on how to do these with a good method would be greatly appreciated:
x^2 + x + 3 = 0(mod 5)
x^2 + 2x + 3 = 0(mod 5)
x^2 = 4(mod 15)
x^2 + x + 4 = 0(mod 8)
x^3 + 2x^2 + 5x + 6 = 0(mod 11)
As I said, I already know the answers from brute forcing and checking the back of the book, but I can not seem to find a consistent, better method, especially for number 5, and this is something I really want to understand how to do.