This question comes up a lot, and I see a lot of bad advice given. This is a the proper way to restore plastic, the Body Shop way. If you perfer the rubbing with blue jeans, and mop n glo way, your looking in the wrong spot.
Now first off, I dont suggest grabbing your lightly scratched discontinued plastic to start with. Find some old ratty plastic to learn on, and practice on. My instructions are going to be detailed, and if you cut corners, it will not turn out good. Follow these to a tee, and your plastic will turn out like new.
Second, Im using plastics that are way beyond repair. These have very deep scratches, that would take days to sand out, and then your plastic would be wavy and not look good once its polished. I purposly left the edges rough, to show progress of the different grits of paper. I also purposly left a few deep scratches to show how a few over sights can ruin a good finish.
Here is the before pic. Notice the deep road rash on the corners of each side. I will be working on the bottom left corner only.