To get the them to sponsor you, hou had to infiltrate the Black Vulkars' base and steal back the prototype swoop accelerator that they had stolen from the Hidden Beks originally. The Hidden Beks were the slightly more responsible and older gang, and offered to sponsor you in the swoop race, and allow you to have the grand prize (Bastilla) if you won, provided you helped them first. The Black Vulkars were vicious, young criminals, and were taking over much of the lower city. The way end up rescuing Bastilla is by winning her in a swoop competition between swoop gangs. You're ultimate objective is to rescue Bastilla and escape the Sith occupied world. The first planet you visit in 'Knights of the Old Republic' is Taris.
