Ace of Carapace - Lestallum
First Available: Chapter 3
Other Requirements: Complete A Stroll for Two
Location: Lestallum - Weapon Merchant
Rewards: 1,500 EXP & Discount at Arms Vendor
In order to accept this quest from the Arms Vendor in Lestallum you'll need to first complete the story quest, "A Stroll for Two" in the city. Once you've completed this quest, all of the available side quests in Lestallum become available. Note: If you're having trouble accepting the quest from this NPC, you actually have to interact with the counter in front of him and not the NPC itself to accept/turn in these quests.
For this quest what you need to do is travel to the location marked on your map and defeat the Bulette enemies here. The location it takes you to is an antenna of some kind and the Bulettes are "trapped" in the small compound. This area is shown in my screen shot below.
Completing this quest will lead to the follow up quest from the same NPC, Tails Spin. This quest is basically the same as Ace of Carapace, just it takes place in a different location.