Why won't a new tri-color cartridge print red?

If what you're saying is still the result after cleaning everything up, there must already be something wrong with the printhead because the printheads are the one's that are responsible for the mixture of the ink cartridges.
Try calling HP Tech Support. They might still be able to replace the printheads for you.

Black printer ink in HP Printer may fail to be recognized by the printer. Most times an error message will be sent indicating that something is wrong with the black print cartridge. Sometimes the printer may accept to print with the black ink, but the printouts may have horizontal lines and very illegible letters. In such a case a manual cleaning of the print head should be carried out. In doing the manual cleaning, one should spray Windex heavily on paper towel and wipe off the surface of the print head.

http://en.kioskea.net/forum/affich-40171-printer-not-printing-black

Have you check your color cartridge if they are properly installed.

Make sure your are not using a refill cartridge.

Replace your tri-color cartridge with original an HP tri-color cartridge.

did you cleaned golden contacts on print head and cartridges with alcohol? if no, try it, other option is, that new cartridge is bad, I had similar problem, bad shades of colors, replaced cartridges and it works now

Are the Jets clean and unobstructed? Depending on printer model, have you cleaned your print heads? Consider also it may be a defective cartridge.

when you change all new cartridges (make sure all cartridge ok, and the cartridge pins connection ok) still red color lost, it is the trouble at print head ink chanel some of that is too dry for longer time less using , most CANON printer head is separated from the ink cartridge, two way you may try:
1) take out the ink cartridge, drop down a little the special cleaner (best clean tools buy from dealer), to the head ink chanel hole, then set back the ink cartridge and waiting minutes
2) try put the print head bottom surface touch a little cleaner or water 1-2 hrs, if you can take out the print head unit. still not fixed for some case that should change the print head unit.

Probably bad, there's actually no way to check ink levels.

Sounds like color cartridge ran out. Were u using color to print black?

Try new color cartridge.

Then if you don't need color go in and set to use black cartridge to print black (text).

If you need color on something only takes a minute to turn color back on.

Since printer worked you can go in and select black which uses the black cartridge and if it prints gives validity to a bad/empty color cartridge.

Good luck.

JC

Yes, this printer has fixed printheads and must have ink in each color to prevent damage to the printheads. Even when printing black only each color is spit in the servicing area to keep the printhead unclogged.

Regards,
Bob Headrick, Microsoft MVP Printing/Imaging

remove print head assembly. clean off.
print head not on cartridge.