If you multiply everything out and group terms you get:

Ax^3 + (A+B+4C)x^2 +(-2A+2B)x +(B+3C) = 3x^3 -16x-1

So there are four equations in three unknowns:

a) A=3

b)A+B+4C = 0

c)-2A+2B = -16

d)B+3C = -1

From (a) you have A=3, then from (c) B=-5, then from (d) C=4/3. Finally, check that these values work in equation (b), but that gives 3-5+16/3 = 0, which is incorrect. So the answer is: there is no solution that works for all values of x. Are you sure you wrote the equation correctly?