See picture.
There are lots of strategies for simplifying expressions. At this point, I do it by rote. When I first learned, though, I made a lot of careless mistakes. But, with practice, it became second nature. So, I would recommend finding an intermediate algebra book (any book used for an Algebra II class) and practicing the problems. If you do even one problem a day for a few weeks, I suspect you will see a marked improvement in your algebra skills.