Trim any loose fibers and irregular edges away from the hole that the dog chewed, using a pair of scissors or a utility knife. This makes the repair much easier.
Trace a template for your patch by laying a piece of paper over the hole and tracing around the hole.
Cut out the template with a pair of scissors.
Use the template to cut out a patch by laying it over a scrap piece of carpet or carpet in an inconspicuous location and cutting around the template with a utility knife.
Apply carpet adhesive to both the bottom of the patch of the patch and the floor in the hole.
Press the patch into the hole.
Place something heavy on top of the patch overnight.
Smooth the fibers so that the repair job blends in with the rest of the carpet.