Answer
$$\color{blue}{ f(x)=x^4-6x^3+10x^2+2x-15 }$$
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 $ 3 $ is a zero, then
$n +3 =0$
$n = -3 $
So our factor is $\bf{(x -3 )}$ because when $x= 3 $, $(x -3 )=0$
If $ -1 $ is a zero, then
$n -1 =0$
$n = 1 $
So our factor is $\bf{(x +1 )}$ because when $x= -1 $, $(x+1 )=0$
So our function is:
$f(x)=(x -2+i )(x-2 -i )(x-3)(x+1) $
$f(x)=(x^2-4x+4-i^2)(x-3)(x+1) $
Recall that $i^2=-1$
$f(x)=(x^2-4x+4-(-1))(x-3)(x+1) $
$f(x)=(x^2-4x+5)(x-3)(x+1) $
$f(x)=(x^3-7x^2+17x-15)(x+1) $
$$\color{blue}{ f(x)=x^4-6x^3+10x^2+2x-15 }$$
