This page explains how to complete the level 86 Endwalker quest "Petalouda Hunt".
Requirements
This quest is available if you have completed the quest Hope Upon a Flower.
Starting the Quest
Talk to Hythlodaeus in Elpis to begin the quest.
Speak with Hythlodaeus
Go northwest to where the orange arrow is pointing in the minimap. Talk to Hythlodaeus to proceed.
Use the Aetheric Rope to Capture Weakened Petaloudai
Go to one of the quest icons on the map. Weaken, but don't kill, a Petalouda. When its health is low, make sure you are targeting it, and use the Aetheric Rope on it from the Key Items section of your inventory, or you can target it and use the icon to the right of the quest name in your list of active quests.
Deliver the Petaloudai
Talk to Hythlodaeus to hand over the two Captured Petalouda. There will be a cutscene. You will receive some glamour gear that you can optionally change into. You can return to the Crystarium for free by going southeast into the building and checking on the Exit to the Ocular in the south wall. Then you can return here by going to the Ocular and checking on the Entrance in the north wall.
Next Main Scenario Quest
After you complete the quest, talk to Hythlodaeus to accept the next quest: In Search of Hermes.