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If the patch is to be applied against a wall and the room has baseboard molding, your two options are: a) remove the baseboard and reinstall it after laying down the carpet patch, or b) leave the baseboard in place and install the tack strip 1/4" away from the baseboard as described in the next step.
If the area you're patching is up against a wall, you'll need to put down a tack strip which has "teeth" that hold the carpet in place. Nail the tack strip along the wall 1/4" away from the edge of the wall or baseboard.
Cut a piece of the foam carpet pad to fit the area to be patched. The pad should be cut to fit inside the tack strip, not on top of it.
Next, check the carpet nap to make sure it follows the direction of the existing carpet. Run your hand back and forth over both pieces to check.
Once you have determined the nap direction for the scrap, use a utility knife to cut a piece to size and fit it into the area.
If one of the edges of your patch is against a wall, attach the other three edges first using the process described in this step. The last edge will be attached to the tack strip in the next step of the project.
Lay down the seam tape just under the seam and use the carpet iron (Image 1) to activate the seam tape and bond the edges together. Then, use a carpet tractor (Image 2) to press the carpet down onto the seam tape to reinforce the bond. Afterward, position something heavy over the area to help set the seam. Continue around the other edges of the patch in the same manner, except in the case where an edge meets a wall as mentioned above.
When all the seams are secure, use a knee kicker to stretch the carpet onto the tack strip. Always work down the length the wall to the innermost corner. Use a hammer to push the carpet down onto the teeth of the tack strip.