Final answer:
To factor the given polynomial function, use the Conjugate Roots Theorem and divide the polynomial by the factors obtained from the zeros. The fully factored form of the polynomial function is

Step-by-step explanation:
To factor the polynomial function
, we start by using the Conjugate Roots Theorem.
Since -4i is a zero, we know that 4i is also a zero.
So the factors are (x + 4i) and (x - 4i). We can write them as (x - 4i)(x + 4i).
Next, we can divide the polynomial f(x) by (x - 4i)(x + 4i) to obtain the remaining factor.
Using long division or synthetic division, we find that the remaining factor is

Therefore, the fully factored form of the polynomial function is
