Verified answer

To do this problem you need to do long division or synthetic division of polynomials. Which does not display well in this format so see references.

You can sort of stumble on the first (x-2) factor by trying f(2) and observing that it equals zero. This implies that (x-2) must be a factor so perform the long division

(x-2) | x^4 - 2x³ - 8x + 16

You get a quotient of x³ - 8

There is clearly a zero at

x = cuberoot(8) = 2

This gives you the second (x-2) factor.

Doing the long division:

(x-2)| x³ - 8

gives you the quadratic factor as a quotient.