Cut out a circular section of carpet around the damaged area. Use light pressure on the utility knife so you cut out only the carpet, not the pad beneath it.
Place the section you removed over your carpet remnant. Match the pattern exactly if you’re working with patterned carpet. Cut around the removed section, using it as a template, to create a size-matched patch.
Return to the hole you cut in your carpet, and lightly brush it so the pile all faces the same direction. Brush your carpet patch until the pile looks similar, and then place it inside the hole. Brush everything together to make sure the pile matches. Gently remove the patch of carpet without upsetting your brush work.
Remove both sides of the double-stick carpet tape, and attach it to the underside of your carpet patch. Re-fit the carpet match into the open section.