The Racer

You can find Firtoril, NPC quest giver, just north of a nearby Skywatch harbor. She stands at the end of an interesting, small, frog racer track.

The first step: Grab a frog. There are couple of nearby Citizens from whom you can get the frog from. Some of them want money for their frog. Others require the Persuasion/Intimidation skills from the Fighter/Mage Guild skill lines.
In order to unlock Fighters/Mages Guild talents you need to complete their introduction spells in Vulkhel Guard. For Fighters Guild introduction mission name is "Aicessar’s Invitation", for Mages Guild introduction mission name is "Curinure’s Invitation".
Unlocking one of these passive skills makes you able to influence some of the citizens without spending gold.

The second step: Examine / Feed the Frog.Next to NPC Firtoril (the one that gave you the quest in the first place), there is a Bag of Frog Chow. This is a small sack next to the end of an improvised frog raceing track. When you get the frog, use this Bag to examine and feed it. Feed your frog ONCE. You can feed it more than once, but in this case the frog is going to be too fat to win.

The third step: Win the Race!Bring your frog near to starting baskets at the begging of an improvised frog racing track. Option to place the frog inside of it becomes available at the start of any race. Once you are able to place the frog inside of the basked, do it. Frog might not win the first time, and you may have to repeat the process more than once.

The last step: Claim your prize!Once your cute frog wins you are ready to complete the quest. This is one of the most interesting and quickest quests out there.

Note:You can not win by shooting other frogs that race against yours. Believe me, I’ve tried!