Apply a cleaning product such as toothpaste or Brasso to a soft cloth. If using toothpaste, make sure it is a paste and not a gel.
Rub this cleaning product into the deep scratches on the CD. The grainy texture of the toothpaste or Brasso will slowly smooth out the CD's surface.
Continue to rub the CD with this cloth while applying pressure until all of the scratches are buffed out. Add more toothpaste or Brasso as needed.
Pour some rubbing alcohol onto another clean, soft cloth.
Rub the rubbing alcohol over the CDs to clean away any remaining residue.
Wipe from the center of the CD to the outside edge with this cloth containing rubbing alcohol.
Let the CD air dry before using it. (See References 1).