Carth's Son Side Quest - Korriban (Kotor 1)
Start Location: Any 'Event Location' within the Docking Bays
Cut Off Point: When you finish the Korriban Story
In Kotor 1 almost all of your Companions get their own "Companion Quest" which involves a story line unique to them. To trigger these quests you will need to talk with your party members and listen to their stories for new Journal entries. Sometimes you can just talk to them on the Ebon Hawk or if you have them with you in the world to advance their personal plot, other times you'll need specific events.
You can trigger events with your Companions on most planets in the port of the city - Tatooine Docking Bay and Dantooine Jedi Enclave are good places to trigger events. Carth's Side Quest can be started anywhere except Korriban and you'll have to finish the Leviathan before you can be approached by Jordo (pictured above).
Jordo tells Carth that he saw his son Dustil on Korriban and he has joined the Sith Academy. As far as I am aware if you've finished Korriban before starting this Side Quest and you turned the entire Sith Academy against you at the end of the main quest on that planet... You'll be unable to complete this quest.
You'll find Dustil in the western wing of the Sith Academy standing next to his bunk (pictured above). Unless you have Carth in your party you will not be able to continue with the second half of this quest. Dustil is as much of a hot head as his father and you'll need to calm him down during your conversation with him to prevent violence.
Warning If you kill Carth's son for any reason during this quest your romance with him ends permanently.
Dustil tells Carth that if he wants to convince him the Sith are evil he needs evidence. You'll find the evidence you need to convince him hidden inside Uthar's room. To get into Uthar's room you can either use a character with high Security skill (14+ gets you in) or you can begin The Double Cross quest to get the key to Uthar's room.
If you plan to go The Double Cross route then you'll need to tell Uthar that Yuthura plans to kill him during the final test. After telling Uthar this you will be able to return to Yuthura and basically rat yourself out by telling Yuthura Uthar knows of her plan.... because you told him.
Yuthura will not be happy but she'll give you the passcard to Uthar's room so you can poison him - which is all we need. There is a Datapad found in a footlocker in Uthar's room which contains information about what they did to Dustil's friend that joined the Sith Academy with him. Carth will acknowledge the Datapad when you pick it up - assuming you have him in your party.
Return to Dustil and show him this Datapad to complete the quest.