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Once you've triggered this quest during the Wherefore Art Thou, Mercurio?
quest, you'll want to leave Mercurio's apartment building and head down the street to your left toward the Medical Clinic. If this is your first time entering the building, then you'll be approached by a fanatic ghoul named Knox. He's not much fun to talk to, but he'll provide you with some information on how exactly a human becomes a ghoul.
The Medical Clinic has two entrances, one in the very front and another in the alley around the back. Depending on your clan or preference, you may want to take the back entrance to avoid any unwanted sightings (especially for the Nosferatu). If you choose the front door, then the clinic receptionist will demand a reason why you're just walking in. If your negotiation skills are high enough, you can tell her that you're either geting some personal things or working on the network to score the key to the second floor.
There are several areas of interest in the Medical Clinic, but one you definitely don't want to miss is in Room 1 here on the first floor. A wounded woman lies on an examining table covered in blood. Feed her your vampiric blood for a point of Humanity and to unlock future encounters with her. She's responsible for the most powerful armor in the game, so I highly recommend paying her this visit.
Continuing down the hallway here on the first floor, you'll find a room marked "Dr. Malcolm St. Martin." Getting inside this room is your primary objective for this quest. There are a couple different ways of getting into this room, but the easiest is probably to use a lockpick (obtained from the game's tutorial or purchased from Trip's Pawnshop). You'll need a rank of 4 in Lockpick, though, so you'll most likely have to use Blood Buff to get in at this point in the game. There is also a key to the office, but it's a little tricky to obtain.
To get Malcolm's Office Key, head to the second floor of the clinic and into the Security room. Here, you'll find a Maintenace Room Key, which will (obviously) get you into the room marked "Maintenance." Now, climb up into the air ducts and drop down the tunnel to the right. Open the next duct opening you come across to drop into a previously inaccessible room that contains Malcolm's Office Key.
Regardless of which way you used to get inside, grab the three Morphine Bottles
on Malcolm's desk and bring them back to Mercurio for a point of Humanity and a point of Experience.
If your character is a Malkavian
... you can use Dementation in your conversation with the receptionist at the front desk, which is an alternate way to get the key to the second floor.
If your character is a Ventrue
... you can use Dominate in your conversation with the receptionist at the front desk, which is an alternate way to get the key to the second floor.