Answer
$$\color{blue}{ f(x)= x^4-16x^3+98x^2-240x+225 }$$
Work Step by Step
We are given zeros and asked to write a polynomial function.
If $2-i $ is a zero, then
$n +2-i =0$
$n -i = -2 $
$n= -2+i $
So our factor is $\bf{(x-2+i)}$
because when $x=2-i $, $(x-2+i )=0$
Since $2-i $ is a complex number, its conjugate, $2+i $ is also a zero
$n +2+i =0$
$n +i = -2 $
$n=-2-i $
So our factor is $\bf{(x -2-i )}$
because when $x= 2+i $, $(x-2-i )=0$
If $ 6-3i $ is a zero, then
$n +6-3i =0$
$n -3i = -6 $
$n= -6+3i $
So our factor is $\bf{(x-6+3i )}$
because when $x=6-3i $, $(x-6+3i )=0$
Since $6-3i $ is a complex number, its conjugate, $6+3i $ is also a zero
$n+6+3i =0$
$n+3i = -6 $
$n= -6-3i $
So our factor is $\bf{(x -6-3i )}$
because when $x=6+3i$, $(x -6-3i )=0$
So our function is:
$f(x)=(x-2+i )(x-2-i )(x-6+3i )(x -6-3i )$
$f(x)=(x^2-4x+5)(x-6+3i )(x -6-3i )$
$f(x)=(x^2-4x+5)(x^2-12x+36-9i^2)$
Recall that $i^2=-1$
$f(x)=(x^2-4x+5)(x^2-12x+36-9(-1))$
$f(x)=(x^2-4x+5)(x^2-12x+36+9)$
$f(x)=(x^2-4x+5)(x^2-12x+45)$
$$\color{blue}{ f(x)= x^4-16x^3+98x^2-240x+225 }$$
