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To brush your teeth, you'll need a toothbrush. In order to get a toothbrush, you'll need to finish a quest chain that only starts after you've completed Eva's quests and obtained a Festival of the Lost mask.
Speak to Eva in the Tower and complete her quests
Complete a run of the Haunted Forest
Obtain one of the Festival of the Lost Masks (has icon of Master Rahool)
Speak to Eris Morn, located at the Moon
In order to start the quest chain, you need to have your Mask equipped.
You should receive the quest item, Box of Raisins
Speak to Spider, located at the Tangled Shore
Complete the quest Even Handed, which requires 12 Fallen, Scorn, Hive and Cabal enemy kills on the Tangled Shore
You should receive the quest item, Winged Chew
Speak to Anastasia Bray, located on on Mars
Complete the quest Armed to the Teeth, which requires 10 Kinetic, Energy, and Power weapon kills on enemies in Mars
You should receive the quest item, Unchocolate
Speak to Asher Mir, located on Io
Complete the quest Precisely, which requires 10 precision final blows on both Vex and Taken on Io
You should receive the quest item, Piece of Celery
Speak to Brother Vance, located on Mercury
Complete the quest In Your Element, which requires 10 Arc, Solar, and Void kills in Mercury
You should receive the quest item, Splice Drops
Speak to Failsafe, located on Nessus
Complete the quest Touched by the Light, which requires 5 Fallen, Vex, and Cabal kills on Nessus using abilities.
You should receive the quest item, Salted Sweets
Speak to Devrim Kay, located at the EDZ
Complete the quest Foregone Conclusions, which requires 5 Fallen, Cabal, and Taken finishers in the EDZ
You should receive the quest item, Sour Engram
Speak to Petra Venj, located in the Dreaming City
Complete the quest The Best of My Abilities, which requires 5 melee and grenade ability kills, and 10 super kills in the Dreaming City
You should receive the quest item, Sugary Engram
Speak to Sloane, located on Titan
Complete the quest Not In Our House, which requires an undisclosed amount of kills in the Solarium, Festering Halls, and Arboretum in Titan
Each kill seems to reward 3.5% progress in the Solarium, and 7% in the Festering Halls and Arboretum
You should receive the quest item, Void Fizz
Return back to the Moon and speak to Eris Morn
Complete the quest The Dance of My People, by dancing for Eris.
You should receive a Goody Bag, and a Toothbrush.
Open your inventory, and use the Toothbrush located in the Consumables section, and you will have finally been granted the ability to brush your Guardian's teeth! If you've finished all the quest steps correctly, you will now have a Used Toothbrush sitting in your inventory.