What is the function with roots 2, 4 + i and 4 - i

I am not sure if you want a function with the roots being
only 2, 4 + i and 4 - i or a function
which has the roots 2, 4 + i and 4 - i.

As there can be an
infinite number of functions that meet the latter condition, I am providing a function
that has only 2, 4 + i and 4 - i as the roots.

f(x) = (x -
2)(x - (4 + i))(x - (4 - i))

=> f(x) = (x - 2)(x - 4
- i)(x - 4 + i)

=> f(x) = (x - 2)((x - 4)^2 -
i^2)

use i^2 = -1

=>
f(x) = (x - 2)((x - 4)^2 + 1)

open the
brackets

=> f(x) = (x - 2)(x^2 + 16 - 8x +
1)

=> f(x) = (x - 2)(x^2 - 8x +
17)

=> f(x) = x^3 - 8x^2 + 17x - 2x^2 + 16x -
34

=> f(x) = x^3 - 10x^2 + 33x -
34

The function with the roots 2, 4 + i and 4
- i is f(x) = x^3 - 10x^2 + 33x - 34