How to get this quest
Complete Lost Engagement Ring and enter the Adventurers Guild in Trademond with at least 5 Infamy.
After seeing Rosemond's guard, go see Rosemond and accept his request by following one of the two possible paths he suggests (seduction or slander). You can postpone his proposal and come back later to accept.
The seduction path
Rosemond will give you successive tasks on this path. Each lewd action on this path ends the day.
The slander path
Rosemond will give you successive tasks on this path, and no lewd action is involved. Instead, 3 Vice actions are taken. Those actions end the days and their minimum base Vice is 10 (requirement you necassarily pass because you could accept the quest).