To start this side quest, speak to the girl in Laundry Room on the first floor.
Once you do, enter apartment 1A where the two old folks are.
Look at the gardening tools in their closet.
Look at the Newspaper clipping on their kitchen table.
Get the neighbour's note on small table by door.
Possess the old woman at the window and choose the "Mysterious Murder" to jog her memory.
Possess the old man sitting in the chair and choose the "Mysterious Murder" to jog his memory.
To solve the events of the murder, choose "the old woman's secret", "the old man's secret", and "a mysterious murder" to put all the pieces together. Return to the Laundry Room and speak to the girl to give her the news and to end the quest.