Gum bleeds because it is swellen, and while brushing your teeth you put pressor on swellen gums, that's why it bleeds.
So first get your teeth scaling done first.
For tooth whitening you can go for progression teeth whitening it dental clinic.
Mix several teaspoons of baking soda with enough fresh lemon juice (or water) to make a paste. Wipe your teeth and any extra saliva off of them with a paper towel. Put a good amount of paste onto your toothbrush and apply. Leave the paste on for 1 minute, then rinse, so as to avoid the acid effecting enamel.
Take care not to brush too careless which injures gums and cause bleeding as in your case
I can help you in this matter with Homoeopathic treatment for bleeding gums. Use heclalava lava toothpaste. Don't brush hard and more than 3 minutes. Brush upword and downword motion. For Medicine I need more details. You can consult me through Lybrate for homoeopathic treatment and further guidance
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