A polynomial with roots A, B, and C can be written as it follows:
[tex]f(x)=(x-A)(x-B)(x-C)[/tex]
Because every time a complex number is the root of a polynomial, it's conjugated also is, let the roots A, B, and C stands for -2i, +2i, and 3, respectively. So, we can write the polynomial as follows:
[tex]f(x)=(x-A)^{}(x-B)(x-C)=(x-(-2i))(x-2i)^{}(x-3)[/tex]
Now, we just need to make the calculation needed to reach the expanded form:
[tex]\begin{gathered} f(x)=(x+2i)(x-2i)(x-3) \\ =(x^2-(2i)^2)(x-3)=(x^2+4)(x-3) \\ =x^3-3x^2+4x-12 \end{gathered}[/tex]
The answer to the present question is the following polynomial:
[tex]x^3-3x^2+4x-12[/tex]
