How do you find a cubic equation with integral coefficients that has the roots 2 and 4 plus i?

If it has integral coefficients and 4+i is a root then its conjugate, 4-i must also be a root.

So the equation is f(x) = (x-2)*(x-4-i)*(x-4+i) where each factor is x minus a root. Then multiply these out.

= (x-2)*(x2 - 8x + 17) = x3 - 2x2 - 8x2 + 16x + 17x - 34 = x3 - 10x2 + 33x - 34