Popitar, I'm not a teacher, I don't even have a major in mathematics, I'm just a curious guy, so you have to check all this stuff with your teacher.
Anyway, I think you are having trouble to understand the basic concepts of proof, but don't worry, you'll get it soon enough (it is quite confusing at the beginning).
You have to understand the binomial theorem, its expansion, to see that you'll have all the expression being divisible by a=x-1. So if a valid substitution yields a multiple of x-1 you're done in your demonstration that x-1 divide the expression, as required.
Now I think is a good idea turn out the internet and get pencil and paper... try to figure out some stuff by yourself now! Use a calculator to verify some ideas, you need to study more the basics, the euclidian algorithm is the most important thing right now.
That's the stuff you must study now in order to advance.