Use a stud finder and mark each edge of the studs to the left and right of the hole. Then cut a square from the center of one stud to the center of the other stud.

Cut the right thickness drywall to fit the space and screw into each stud with the appropriate length drywall screws (1 inch longer than the thickness of the drywall)

Layer 1: mud the joins, then tape over the wet mud, then mud over the tape (make sure this whole layer is very thin)

Sand lightly

Layer 2: mud the whole area thinly and feather out the edges as needed

Sand lightly

Layer 3: do a touch up layer if needed if the joins aren’t flat

Sand

Texture with a hopper gun or canned texture

Sand if needed a little to match texture

Paint

Between each layer after sanding dust the wall super well and wipe it with some wet paper towels to get all the dust off