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Lean a piece of drywall horizontally against the wall in the area where you plan to install it. Drive long screws into the studs closest to the edges of the drywall, 48" from the ceiling (Image 1). Leave about 1" of the screws protruding from the studs so they can support the drywall.
Rest the drywall on the screws (Image 2). You'll probably need some help to lift it.
Using a framing square and a pencil, draw light lines on the drywall to indicate where the studs are located. This will help when you start driving the screws.
Secure the drywall to the studs with drywall screws. The screw head should go just below the surface of the paper. Be careful to drive the screws only enough to dimple the surface; it's easy to overtighten and cause the screw head to go all the way through the paper surface. If you accidentally break through the surface, drive another screw nearby.
Remove the supporting screws you installed earlier. Then lean the bottom piece of drywall against the wall, and transfer the stud lines onto it. You might want to use a speed square or a level and a straightedge to help you draw the lines.
Jack up the bottom of the lower piece of drywall so it butts closely against the upper piece. If you don't have a helper, use a drywall jack (Image 1) to lift the lower piece. It's okay if there's a slight gap between the top and bottom pieces.
Use drywall screws to begin securing the lower piece to the wall studs. Drive the first screws through the beveled edge at the top of the drywall (Image 2).
After the drywall is in place, apply drywall mesh tape along the joints between the two pieces of drywall.
Use a wide drywall knife to spread joint compound over the mesh tape. Don't try to fill the entire gap all at once; apply a little at a time, and allow it to dry before applying a second coat. Then, cover the screw heads with drywall compound. You don't have to cover each screw head separately; instead, you can speed things up by spreading the compound down each column of screw heads.
Allow the joint compound to dry and then sand it smooth. Regular sandpaper works fine, but you might get faster results by using a drywall sanding screen like the one shown in the image. Wear breathing protection during this step because the process generates a lot of dust.
Repeat the process as necessary until the wall is perfectly smooth, then prime and paint the walls.