Pull up the strips and buy some new ones. Coming out of concrete they will most likely break. Besides they don't seem to be in good shape.
Put the new strips in and maybe add a bead of urethane adhesive under them to help secure them. If they don't nail in securely, a second row of tackstrip can be added. Keep about a 3/8" gap between the tackstrip and the wall because you need a place to tuck the trimmed carpet edge down into when finishing up. If you haven't nailed tackstrip into concrete before, you might (will) find this a bit difficult. Tackstrip can just be glued if you use a good adhesive. Cut the tackstrip into 8 or 10 inch pieces put a 1/8" or 1/4" bead of adhesive on each piece and press them into place. Full length strips are often bowed and won't lay flat.... short pieces will.
Once the tackstrip is in, (be sure to point the tacks towards the wall), fit the new padding and use duct tape to cover the pad seams.
Now fold the carpet way back into the room so it lays flat. Trim all 3 sides of the carpet about 1 inch back from the existing edge. The edges needs to be trimmed with a sharp razor knife (new blades) so you have three perfectly straight edges that are cleanly cut, not ragged.
Now you are going to have to trim your fill piece to fit into that shape. Use a large enough fill piece so that you can trim the edges more than once............ I'm doubting you will have success first attempt at matching all three sides. You might try poster board and make a pattern of the shape and transfer that onto the back of the fill piece.
Oh, and be sure the fill piece lays the same direction as the carpet that's already down.
Seaming the fill onto the main carpet requires hot melt tape and a carpet seaming iron. That sticky tape for carpet will not work.
Carpet seam edges are supposed to be sealed with a carpet seam sealer prior to seaming to keep the seam from unraveling over time. Not many pros do this, but it's supposed to be done.
Not sure of your abilities, but maybe you could get the old strip removed, new strip in place and pad installed, then have the fill piece of carpet laying in the room and have a pro do the seam and stretch it back in.
If you want to do it all, carpet seam irons can be rented at some retailers or home centers.
There are how to's on Youtube about using seamers.
....just don't use a sticky type of carpet tape, no matter how sophisticated it looks.
