How many integer solutions of $217x^3+8x^2+15x+15=0 \pmod {77}$?

Question

How many integer solutions of $217x^3+8x^2+15x+15=0 \pmod {77}$?

This is my question. I have already done $f(2^8) = 0 \pmod {7}$, but I can't solve this thing.

Answer 1

The polynomial modulo $7$ becomes $$ x^2+x+1=x^2-6x+8=(x-2)(x-4) $$

Modulo $11$ the polynomial becomes $$ 8x^3+8x^2+4x+4=4(x+1)(2x^2+1)=8(x+1)(x^2+6)=8(x-10)(x-4)(x-7) $$

Can you finish?