You should install the carpet with a separate piece for each step, not one whole piece for the entire stairs. First install tack strips on the sides of the stairs and at the back of each tread. Measure each carpet piece to cover from the back of the stair tread to the bottom of the riser. Cut the pieces about ¼” larger than your measurement to leave room for padding to fit underneath. Wrap the piece around the step, starting from the bottom of the stairs. Tuck the bottom edge into the carpet of the previous stair (or the room itself for the first step). Use a hand stretcher to then tap the carpet into place unto the tack strips.
