To complete Professor Garlick's Assignment 1 in Hogwarts Legacy, you need to purchase the Venomous Tentacula from the Dogweed and Deathcap plant shop, equip and cast the Venomous Tentacula, and also complete the task involving the Mandrake. Afterward, attend Herbology Class and then report back to Professor Garlick to inform her of the completed tasks.