Location: The Hideaway (ship)
Petitioner: Nigel, Head Botanist
Unlocked: Complete Fire and Ice
Reward: Morganbeard Extract, Black Blood, 15 Renown
In order to unlock this side quest you need to first complete Fire and Ice. This is a major, main story quest, and upon returning to the ship Clive will be by himself. If you want a bit of help, complete the short main quest After the Storm, as well, so Jill is back in the party.
The petitioner for The Root of the Problem is Nigel, the lead botanist of the Hideaway. Visit him in the Backyard, the farming area in the north of the ship, and he'll ask Clive to fetch him some Morganbeards. They're a flower that grows near Rosaria, specifically Sorrowise.
Make a trip to either Martha's Rest or the Auldhyl Docks Obelisks - they're about the same distance from your destination - and head into Sorrowise. You're aiming for a stretch of land to the north of Glaidemond Abbey. You'll know you're in the right area when you start running into Goblin Gardeners, which are a higher level than the average for this region.
Arrive in the correct spot and you'll find more Goblin Gardeners, and with them is a Gigas Ploughswain. It fights the same as any other Gigas you've encountered, and shouldn't be a big deal to defeat. Take out the baddies and you'll be free to collect the Morganbeard, which grows in patches of yellow flowers. You need to grab three, each by the edge of the water.
Take the Morganbeard samples back to Nigel at the Hideaway to complete the quest and earn your reward. With the Morganbeard Extract in your inventory most consumables will either restore more health or last longer than normal, making this an excellent item to have indeed.