Squirt some toothpaste or Brasso onto a soft cloth.
Rub the toothpaste or Brasso on the cloth into the deep scratches on your CD. Apply pressure while rubbing. As you do, the grainy texture of the toothpaste or Brasso will wear away the top plastic layer of the CD and smooth out the scratch.
Add more toothpaste or Brasso to the cloth as needed. Continue rubbing the surface of the CD until all of the scratches are gone.
Pour rubbing alcohol onto another clean cloth.
Wipe the surface of your CD with this cloth to clean any remaining toothpaste or Brasso residue.
Allow the CD to air dry before testing it. (See Reference)