The answer to this question is different for every pet. There is no formula that can be used to time dental work in all dogs and cats. I am sorry, but the best answer I can give is: pets need dental work when they need dental work.
So, how can you tell when your pet needs dental work? There are several symptoms of dental disease. Bad breath and discolored teeth are common. As well, pets may become lethargic, their coats may smell bad, they may drool, and red gums or loose teeth may be noted. Most pets with dental disease do not lose their appetites.
Dental disease is serious. It is very painful. It has been linked to kidney problems, cancer, heart problems, premature death, diabetes, bladder infections, tooth loss, and many, many other problems.
Vets do themselves a tremendous disservice when they refer to animal dental work as “teeth cleaning”. Sure, we clean the teeth. But we also do a complete intra-oral exam, probe pockets, perform supra- and subgingival scaling, perform root planing, extract or repair teeth, and perform intra-oral radiographs. In the end, the teeth are clean, which is nice. But the fancy-sounding medical procdures are what really help the pet. They make the mouth healthy.
To get back to your situation: if your dog has red, bleeding gums, she almost certainly needs dental work. At the very least, you should have a vet look at her mouth.
And, although no formula can determine how often pets should undergo dental work, there is a simple rule that you can use to make sure your pet’s mouth stays healthy. Go to a good vet for regular checkups every 6 months. Have the vet assess the teeth (as well as the rest of the body!). The vet will be able to tell you if and when dental work is needed, and whether the benefits of dental work outweigh the risks of anesthesia.