1. ## Hensel's theorem

Find all solutions to the congruence 7x^4 + 9x^3 + 2x + 9 ≡ 0 (mod 3375) .

I was given the hint that there are 8 solutions.

So i've found that 15^3=3375 so we start with mod 15 and so I have f(3), f(4), f(9), and f(13) are all congruent to 0 mod 15 so they are solutions. then we find f'(x) and plug in each solution, but i am getting lost in the process and I feel like I need to see it step by step to unscramble my thoughts.

and also Find all solutions to the congruence 2x^4 − 3x^3 − 9x^2 − x + 10 ≡ 0 (mod 214375) . with the hint that there are 4 solutions.

2. You might find it easier to start with the roots mod 3 (namely 0 and 1) and the roots mod 5 (namely 3 and 4). Use the Hensel procedure on each of these separately to find the solutions mod 3^3 and mod 5^3, then apply the Chinese remainder theorem.

The same method should deal with the other problem, using the fact that $214375 = 5^4\times7^3$.