A good set of luggage can last you a lifetime, but keeping it in good condition isn't always the easiest task. Baggage handlers toss luggage around all day long, and your hard-shell luggage may get a scratch in the process. Fixing the scratch takes a little bit of creativity, but if the scratch isn't too deep, you should be able to get it out with a few basic techniques.
Wet the cloth with water, then rub the wet cloth across the scratches to remove any light surface scratches or oil stains.
Lightly run the eraser across the scratches, working the eraser down the length of the scratch. Wipe off any residue from the eraser using the cloth.
Apply a small amount of toothpaste to the scratch. Work the toothpaste over the scratch with the toothbrush, working the brush in a circular motion. Wipe off the toothpaste using the cloth.
If the scratch is too deep, you have to decide whether repairing it is worth it. Deep scratches require sanding out, then repainting, which gets awfully expensive for a piece of luggage that may just get damaged again. If you don't want to go through that hassle, consider covering up the scratch with a sticker or something similar, which will also help you identify the piece on the baggage carousel the next time you fly.