Which question is to be answered? The polynomial or the complex number problem or both?
p(x)=ax^3+bx^2+cx+d is the polynomial. When x=0 p(0)=8, so d=8. p(x)=0 when x=4, 1, -1, so these must be the roots of the polynomial: (x-4)(x-1)(x+1) which expands to (x-4)(x^2-1) becoming x^3-4x^2-x+4. This needs to be doubled so that p(x)=8 when x=0. The polynomial p(x) is 2x^3-8x^2-2x+8.
More to come...