On this page of the game guide to Borderlands 2 you will find a detailed description of the side mission titled Medical Mystery. You can start this quest in the Sanctuary location after the completion of the side mission named Do No Harm.
1) Zed's clinic
Three Horns Valley
2) Southern entrance to Dr. Mercy's hideout
3) Northern entrance to Dr. Mercy's hideout
4) Dr. Mercy
5) Bandit camp
Zed  will ask you to find the source of the strange wounds. Head to Three Horns Valley. Enter Dr. Mercy's hideout through the southern  or northern entrance . Head towards where Dr. Mercy is , killing Bandits who cross your way. Mercy will be using a shield, so if you're using Explosive ammo, shoot at his legs. A Corrosive type weapon will also do fine. If you don't have such a weapon, you will need to try to approach him from behind.
After defeating Dr. Mercy, approach his corpse to end this mission and begin another one. In your inventory, you will find the Blaster which you should equip. You will now have to kill 20 Bandits by using this weapon. In order not to run out of ammo, try firstly weakening them with other weapons. You should be able to kill enough enemies in this location, though if for some reason you won't make it, head to the nearby Bandit camp  to finish the task. Return to Dr Zed  to finish the mission. In return you will receive a very good pistol with a Fire Elemental Effect.