The argument is simple. If God is all-powerful, he should be able prevent all evil and suffering. If God is good, he should want to. But if God can remove all evil and suffering, and he wants to remove all evil and suffering, then why is there still evil and suffering? Therefore a good, all-powerful God does not exist.
Very nicely put. I think Rom. 9:22-23 addresses the question most directly, just understand that the "What if" is rhetorical. Paul is not speculating but telling us the answer, which in brief is that God wants the cross. If we knew him perfectly but did not know his wrath and mercy we wouldn't really know him and that would be more evil than all the evil in the world. It is therefore good that there is evil.