BudsGunShop.com - +P allowed

David, Does your box actually say +P ? 340 grain bullet isn't anything you can readily get a hold of, is this Buffalo Bore or something like that?
There is no SAAMI specification for a +P 44 magnum load, that I know of. The term 'magnum' indicates longer case, more powder, and higher pressure.
Short story, the M44 should be able to handle any 44 magnum load you can feed it...... ,butIf you are speaking of the Buffalo Bore hard cast 340 grainers, I would caution against using that in an M44. The M44 is nearly as strong as a S&W N-frame and Buffalo Bore only recommends their hottest loads for Ruger, Freedom Arms, and Raging Bulls.
NEW HEAVY 44 MAGNUM +P+
This new load is designed for only certain revolvers that have the cylinder length to handle it. They are as follows. Ruger Red Hawk, Ruger Super Red Hawk, Ruger Super Blackhawk or Vaquero, Freedom Arms Model 83, Taurus Raging Bull and Dan Wesson Revolvers. Suitable rifles include T/C Encore, "modified" Marlin 1894, Winchester 1894, any rifle with a falling block action and the Handi Rifle.

David, for safety and warranty reasons, I would contact Taurus and get their opinion on the specific ammunition you have at Taurus Customer Service : (800) 327-3776 Monday thru Friday* – 8:00 AM to 10:00 PM (EST)