Seal off the vents and cold air returns in the room with painter's tape. This is a dusty project. Failing to seal these vents will result in dust permeating your entire house.
Tape up dropcloths over doors. This will help prevent airborne dust from moving room to room.
Sand down the white line with a belt sander with a medium-grit sanding belt. Start the belt sander in the air. This will give you more control and prevent you from gouging the floor. Lower it onto the first board slowly and begin sanding. Sand with the grain of the wood.
Vacuum all the sawdust in the room. If necessary, wipe down the boards with a tack cloth. Fine pieces of dust will ruin the finish you are about to apply.
Apply matching wood stain over the sanded areas with a paintbrush. Brush with the grain of the wood and apply a very thin coat. Let this first coat dry for four hours. Sand the area lightly with a fine-grit sandpaper. Apply a second coat and allow it to dry. If the color matches, you can move to the next step. If not, keep going, sanding between coats, until you have matched the color of the rest of the wood.
Lightly sand down the newly-stained areas with a fine-grit sandpaper. This will allow the polyurethane to create a good bond with the wood.
Apply two coats of clear polyurethane over the newly-sanded areas with a paintbrush. Once again, go with the grain of the wood. Wait for the first coat to dry for at least four to six hours. Again, lightly sand the area. Apply the second coat. Let it dry for eight hours before allowing foot traffic.