James, I've never done this but I've seen my husband and brother-in-law do it often enough to understand the method. You need to cut a piece of drywall the same size as the hole you've cut. Then you'll need the special drywall tape to tape it into the hole going around all 4 sides where the ceiling and patch meet. Spackle the edges and let dry. Sand and respackle and sand until the edges are perfectly smooth. Then you can repaint.
Get a piece of sheet rock cut to size. See the studs already there. Screw the sheet rock into place. That will probably take about 6 screws. Get some sheet rock tape (LOWES< HOME DEPOT) Put sheet rock mud where the new sheet rock meets the old, just a thin coat, then tape. Put mud on top of the tape and screws. Sand. Do the mud again. Sand. Do this again and again until it is perfect. Good luck, its not hard, but time comsuming.
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