From AQHA rule book;

Appendix Horse
A horse registered with AQHA which is the result of breeding a Thoroughbred and an American Quarter Horse that has a permanent number, or a combination of an Appendix numbered American Quarter Horse and an American Quarter Horse with a permanent number. Appendix horses are distinguished by an "X" in front of their registration number and their certificates are gold.

There are three different mixes of horses that are eligible to be registered in AQHA's registry:

Registered Appendix (X) + Registered American Quarter Horse (QH) = Registered Appendix foal(X)

Registered American Quarter Horse (QH) + Registered American Quarter Horse (QH) = Registered American Quarter Horse foal (QH)

Recognized Thoroughbred (TB) + Registered American Quarter Horse (QH) = Registered Appendix foal (X)

Some examples of horses that are not eligible for registration with AQHA:

Appendix (X) + Appendix (X)' not eligible to be registered
(Unless one Appendix horse advances)

Thoroughbred (TB)+ Appendix (X)'not eligible to be registered
(Unless the Appendix horse advances)

American Quarter Horse (QH)+ Any other breed of horse besides a registered Appendix, acknowledged Thoroughbred, or registered American Quarter Horse not eligible to be registered with AQHA