Lay drop cloths down to contain the mess from tearing out the Sheetrock.
Cut out the damaged section of Sheetrock with a utility knife. Cut the Sheetrock all the way to the studs on either side of the damage. It will be much easier to replace a full sheet than to fit in a partial piece. Remove any screws or nails from the studs using a claw hammer.
Measure the open area with a tape measure. If it is not the size of a full piece of Sheetrock, use a utility knife to cut the Sheetrock to the size needed.
Install the new Sheetrock by fastening it to the studs using an electric drill with a driver bit and drywall screws.
Place drywall tape on all the seams of the new piece of Sheetrock. Ensure that the tape is completely flat and does not have any wrinkles.
Fill a drywall pan with joint compound and use a 6-inch drywall knife to spread the joint compound over the taped areas. Ensure that the joint compound is as smooth as possible. Let it dry for 24 hours and sand with a sanding sponge. Make it as smooth as you can.
Apply a second coat of joint compound with a 10-inch drywall knife in the same way as you applied the first coat. Let it dry for 24 hours and sand smooth. Wipe the area with a damp sponge to remove any drywall dust before you paint.