Private Renolds and his squad were ambushed by Caesar's Legion, and some his fellow troopers were taken prisoner. He managed to give his captors the slip, tracking them all the way to Techatticup Mine.
The quest is started by talking to the private up the road from the mine or south from Nelson. Note that it's possible that the game will simply refuse to let the player interact with the prisoners until the quest is active.
The soldiers in question are located inside the mine, down the right-hand corridor. They are in locked storage cells (Average difficulty; Alexus carries the key to the doors) and have to be untied to be freed. The Legion will immediately attack them, so the player will have to clear the way for them.
A silenced weapon and good skills or ammo allow for cleaning the cave out with no loss of reputation with the Legion. Melee weapons are a lot more difficult.
After freeing them, the player needs to report back to the private, who will be relieved if they are freed (or distraught if they were killed). Only one prisoner has to make it out the mine's door.
Rescue the NCR hostages from the Techatticup mine.