Ok, well, I guess I'm going to be a tiny bit lazy and instead of typing up some notes myself, I'll just give a link to some other good websites:
Remember that we are using polynomail long division as the first step in re-writing rational functions (a fraction with a polynomial as numerator and another as denominator). The second step is to break the denominator into even smaller pieces using partial fractions. The overall purpose is to be able to find the integral of any rational function.
Both of these pages include how to do the division as a process, as well as some examples/exercises for you to try out on your own (which you should do for practice before the exam!)