One of the great things about cartridge-style faucets is how easy they are to service, but when hard water deposits collect on the O-rings and the valve housing, what was once easy is now not easy at all. You should be able to pull the cartridge out with pliers after pulling the pin or unscrewing the nut to release it, but when the cartridge is bound by minerals, you often have to resort to generic cartridge pullers. But it's usually better to use one designed for your particular model.
Why the Cartridge Is Stuck
A shower faucet cartridge is usually fitted with at least one O-ring, sometimes more, and it's this rubber part that binds the cartridge. Rubber is especially vulnerable to mineral accumulation, and after the faucet has been in use long enough, the O-rings basically fuse to the cartridge and the valve housing. The plumber who installed the cartridge could have forestalled this by lubricating the O-rings when installing cartridge, but apparently that didn't happen, so here you are.
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Minerals can also bind metal and plastic and that only adds to the problem, especially if the cartridge is old and hasn't been serviced for many years, which is often the case. In extreme cases of scale buildup, you'll often see deposits on the valve stem after you remove the handle and on the exposed rim of the cartridge after you take off the escutcheon.
Using a Cartridge Puller
Sometimes, all you need is a plastic nut that fits over the valve stem and locks onto it, which allows more leverage for rotating the cartridge with a wrench so you can break the bond holding the cartridge. Moen supplies this simple tool with some repair kits for its 1222 shower faucets. If you have one and it doesn't work, you can often improve its effectiveness by heating the cartridge with a hairdryer for 10 to 15 minutes.
If you have Moen faucet, you can usually find a model-specific cartridge puller for that faucet. For example, the Moen 104421 cartridge puller works with 1200, 1222 and 1225 cartridges, and it makes other pullers for different models. You use one of these by locking the base onto the valve housing and screwing the central screw into the hole on the valve stem for the handle. When you turn the nut at the base, the screw pulls on the valve stem and removes the cartridge far enough for you to finish the job of extracting it with pliers.
A more generic type of cartridge puller looks like a screwdriver with a T-handle. The tip of the tool locks onto the valve stem by means of a set screw, and the handle gives you the leverage you need to rotate the cartridge and break the bond, then pull it out.
Cartridge Puller Doesn't Work
It's unlikely a Moen cartridge puller with a screw mechanism won't extract a cartridge, but you can't generate as much force with a generic puller, so you may need help. You can use a hairdryer, but a more effective way to soften and dissolve scale is to inject vinegar into the valve housing using a syringe. Let the vinegar work overnight and, in the morning, you may be able to extract the cartridge even without a cartridge puller.