How do you find a 4th degree polynomial with integral coefficients that has zeros of 2 ( multiplicity of 2) and -i?

You know the roots. 2 occurs twice as root. If -i is a root, so is i.

Therefore, we have

(x-2)^{2}(x+i)(x-i)=0

But (x+i)(x-i)=x^{2}+1

So, the resulting quartic polynomial is

(x-2)^{2}(x^{2}+1)

Expand out to find the polynomial .