The result you have stated is wrong. The correct statement would be something like:

Show that for some polynomial g(x) with integer coefficients.

First note that x = 1 is a root of f(x) and so x - 1 is a factor of f(x). Then .

Now note that x = 1 is also a root of f'(x). Then x - 1 is a factor of .

It follows that (x - 1) is a factor of h(x). Therefore and the result follows.