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for p(x)= x^3− 3x^2+ 27 ≡ 0 (mod 1125), i get that 1125 = (3^2)(5^3).

p(x) ≡ 0 (mod 3^2), p(x) ≡ 0 (mod 5^3).

x ≡ 0, 3, 6 (mod 3^2) for 3^2

for 5^3, x ≡ 51 (mod 5^3)

then i get x=801, 51, 426 (mod 1125).

but i cannot seem to get as eloquent of an answer for p(x)= 4x^4 + 9x^3 - 5x^2 - 21x + 61.

can anyone help? i know you start out the same way. perhaps there is an easier way?