1) Dividing 9a + 3b + c = 43 by 9 gives some restrictions on b and c. Can this even be done?
2) f(0) = - 1 is one root. But it might also be a double root. You need to check this case as well.
I'd suggest a slightly different procedure for your problem. Don't express f(x) = ax^n + .... + d. Use f(x) = a(x - b)(x - c).... and restrict a, b, c,... as integers.