Unscrew the hinges with a power drill and remove the existing door.
Remove the door casing on the interior and exterior sides of the opening by cutting, with your utility knife, through the caulk where the casing meets the doorjamb. Drive a wide putty knife between the casing and the drywall to protect the drywall as you pry the casing off with a curved pry bar.
Expand the door to the appropriate width. Cut through the wallboard on the side of the doorway you want to expand with a rotary cutting tool.
Cut the nails attaching the door frame to the stud with a reciprocating saw on the side of the door you are expanding. Dislodge the frame completely with a hammer.
Remove the stud by cutting through the nails at the bottom of the stud using the reciprocating saw. Pull the stud free.
Cut through the base plate with a Japanese hand saw to prevent damaging the floor. Pry the base plate from the floor using the claw of a hammer.
Nail a wall stud into place at the new width with a nail gun. Nail a jack stud to the king stud with a nail gun to support an extended header above the door.
Extend the length of the header. If you're expanding the opening by 6 inches, attach a 6-inch 2-by-4 to a 12- or 16-inch 2-by-4 using a nail gun with one end lined up. Screw this in place with the longer 2-by-4 resting on top of the existing header and the shorter piece extending to the header to the new wall stud.
Install the top doorjamb by nailing a 36-inch long 1-by-4 to the header.
Hold the door-height 1-by-4 lumber in place against the wall stud. Wedge shims between the lumber and the wall stud every few feet and nail the lumber in place to create the vertical doorjamb. This prevents flexing of the doorframe when the door is opened and closed.
Cut the casing to the correct length and angle with a miter saw. Hold the casing in position and nail it in place around the doorway on both sides of the opening.
Apply joint compound to any remaining visible joints around the door casing with a putty knife. Wet a strip of paper tape and apply it over the joint compound, pressing it firmly into the compound with a putty knife.
Attach a new door either with the old hinges and screws or a new set using a power drill.
Fill any exposed nail holes with joint compound and apply a coat of paint for the finishing touch.