Cultured marble is made of a resin - it's more similar to fiberglass than stone. So to get scratches out of cultured marble you want to buff them out the same way you would on a fiberglass auto body panel. Go to an auto parts store and get the following stuff

- A buffing pad that you can use with an electric drill
- Medium grit rubbing compound that is designed for fiberglass
- Paste wax, also designed for fiberglass

Once you get this stuff home you can use the buffing pad with the rubbing compound to buff out the scratches in your cultured marble. Once the scratches are gone, clean the area with water. Then apply the paste wax to seal the surface.