paint scuffed on underbelly of front bumber

A long time ago my wife made a wrong turn and scraped the under side of her car on a curb, there are two ways to go about it really, it depends on if you are wanting to spend the time and money to fix it the right way or not. If you want to fix it the right way your self, check online and figure out your car color and order some of the paint and some plastic bonding clear coat of your choice, you'll want to sand down the damage area until smooth and then carefully make light overlapping coats (make sure to wait between coats) and then do the same process with the clear coat. Make sure to buff the bumper. OR if you don't want to go into that much detail since it is under the car and not visible you could always just order the color matching paint and put some on to cover the scratches.

This is very helpful Matt thank you.....can I sand this by hand and if so can you recommend a sand paper product?

When I re painted my bumper I used an electric one, I used one of those mouse ones. It depends on how deep it is, if it isn't deep I'd suggest using a really high grit like 1000 to start and keep going up until its smooth and you can't feel any scratches/scrapes if its pretty deep start with a lower grit, it all really depends on how deep it is

Repainting your BMW with a can of spray paint is the surest way to devalue your car. I would bite the bullet and have a pro do the work and then look into one of these clear film protectors that protect paint in vulnerable areas.