Given that 4 - $2 i$ is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable.
\[
f(x)=x^{4}-11 x^{3}+34 x^{2}+20 x-200
\]
Answer
\[
f(x)=
\]

Step 1 ：The Conjugate Roots Theorem states that if a polynomial has real coefficients and a complex number a + bi is a root of the polynomial, then its conjugate a - bi is also a root. Since we know that 4 - 2i is a root, we also know that 4 + 2i is a root. We can use these roots to factor the polynomial.

Step 2 ：The polynomial has been factored into two parts: \((x - (4 - 2i))(x - (4 + 2i))\) and \(x^2 - 3x - 10\). We can further factor the second part.

Step 3 ：The polynomial function can be factored completely as \(f(x)=(x - (4 - 2i))(x - (4 + 2i))(x - 5)(x + 2)\)

Step 4 ：\(\boxed{f(x)=(x - (4 - 2i))(x - (4 + 2i))(x - 5)(x + 2)}\)