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Background
At the start of the game you'll receive the quest
A Mysterious Murder, which will task you with solving the murder of Councilor Jake in
Cyseal City. Most people in the city will have an opinion about the matter, and they'll all assume that Jake's wife Esmeralda is the killer. When you actually find evidence against her, you'll trigger this quest.
Walkthrough
There are three pieces of evidence implicating Esmeralda in Jake's murder:
The pieces of evidence will earn you 260 xp each when you pick them up, but when you ask Esmeralda about them, she'll be able to explain them away. Moreover, after showing her all three items, she'll claim that Jake received a "fatal" wound while hunting boars, and that he became obsessed with Evelyn, Master Thelyron's apprentice, when she healed him. Esmeralda will then suggest that Evelyn is a better suspect than she is. You'll earn 200 xp at the end of this conversation.
At this point, you can show the evidence to Captain Aureus in the Legion Headquarters (#3), and you'll get an option (at the top level of the dialogue tree) to have Esmeralda arrested. This will give you a choice between being Righteous (leadership +1) or Renegade (pickpocketing +1). If you decide that Esmeralda should go to jail, then you'll receive 1800 xp, but you'll lose Esmeralda as a shopkeeper for the remainder of the quest.
Eventually, at the end of the quest
A Mysterious Murder, you'll discover that somebody other than Esmeralda killed Jake. If you had Esmeralda arrested, then you can tell Captain Aureus about the crime, and ask him to have Esmeralda released. He'll call you a few names, but he'll immediately head to the prison and let Esmeralda out of her cell. This will earn you 900 xp plus a point of reputation, and Esmeralda will go back to her house and become a shopkeeper again.
1 - Esmeralda's House
2 - King Crab Inn
3 - Legion Headquarters