f(x) = x^5 + 3x^2 + 4

Find all solutions to the congruence f(x) congruent to 0 mod 12 by using the Chinese remainder theorem.

I'm not sure exactly how to do this properly.

I started by creating two congruences since 12 = 4 x 3.

I manually tried entries 0 to 3 for mod 4 and entries 0 to 2 for mod 3.

The only common solution is x=2, so I tried x = 2 as well as x= 5, 8, 11 (because of mod 3) and x= 6, 10 (because of mod 4) in the original polynomial.

My final answer is x = 2, 5, or 8.

1) Is this right?

2) Whether or not it's right, how do I work this problem and present the solution properly?

Thanks in advance.