Clearly you haven't tried enough. You said this problem was posed years ago and now you couldn't spend just a day trying to figure it out with my help without looking up the solution. This is simply bad practice for learning, because you haven't learned how to arrive at the solution, only the solution itself, which is a bit useless.
Anyway, to answer your question for why you should consider the ideal (p,b), the answer is that you want to exploit the PID property. It says that if you have an ideal, it's principal, and you need to exploit this somehow. You have three numbers, p,b and c. You want to build an ideal out of them for which it being principal is non-obvious or not true in general. The way you know how to make ideals is to take a few elements and let the ideal be the smallest ideal containing those elements, i.e. the ideal generated by those elements. And really there's only so many ways to do it, you can do (p,b), (b,c), (p,c), (p,b,c), so pick one of them and see where your nose leads you further.
Now let's try to do the other question instead.>>Prove every PID is a UFD
As another anon suggested, when you're proving properties of PIDs it's always good to first try to prove them for Z, which is the most important model of a PID and the object from which the concept of PID was generalized.
Do you know how to prove that Z is an UFD? Don't look it up, try to derive it on your own. I can help you through if you get stuck by guiding you. Tell me what you come up with.