Home DIY

https://www.youtube.com/watch?v=lj7YK1lIRUM

This video covers the process of cutting the piece fairly well. I'd personally add two strips of wood either side of where your new hole will be.

Once you have screwed the new piece in place, don't fill over the cracks like on the video. Instead, make a 45 degree cut along all the edges with a stanley knife to create a groove all the way around the repair. Now clear any debris or loose pieces left in the groove and with a small paint brush apply a mixture of PVA and water to any of the exposed plaster of the plasterboard. This will seal the plasterboard as otherwise it will suck any moisture out of the filler you use and it will be more likely to crack as the filler won't adhere well. If doing bigger repairs, you can add a bead of gripfill at the bottom of the groove to help prevent cracking.
Some people will tape over the cracks with a scrim tape before filling. This is stronger, but it will require using a lot more filler, and is trickier to get neat if you're not used to filling plasterboard.
Fill over the repair, making sure to push your filler into the groove to remove any air pockets. Try and feather it out as best as you can around the edges too. Be aware that it may be difficult to get a smooth finish on the first fill where the filler is in the grooves.