Exterminator (Fighters)

Rat Hunt is the first quest that Eydis Fire-Eye will give you, once you've joined the Fighters Guild.

Detailed Walkthrough
Talk to Eydis Fire-Eye inside the Balmora Fighters Guild Hall about orders and she will tell you that a local Dunmer named Drarayne Thelas has a cave rat infestation inside her house in Balmora and Eydis tells you to speak with Drarayne, ask her where the rats are located inside her house, kill the rats, and report back to her.

To get to Drarayne's house you have to cross from the western end of the central bridge and enter the house through the front door on the eastern end of the bridge.

When you speak to Drarayne about her rat infestion, she will tell you that she's locked three rats inside her house, one inside her bed room and two inside the storage room up the staircase outside her house. Afterwards, she gives you a key to her bed room and the storage room inside her house and she tells you to kill them, before they chew her very large supply of pillows. After you've killed all of the rats, talk to Drarayne again and she will give you 100 gold as a reward for killing the rats and report back to Eydis to complete the quest.

Quick Walkthrough
1.Speak to Eydis Fire-Eye about orders.

2.Speak to Drarayne Thelas to get the key.

3.Kill the rats.

4. Collect your reward from Drarayne.

5. Report back to Eydis.

Rewards
Speak to Drarayne Thelas after you've killed the rats and receive a 100 gold reward from her and report back to Eydis Fire-Eye to complete this quest and speak to her about you're next orders, starting the Egg Poachers quest.