What is the best way to do carpet patching?

I have had a 5 foot archway opened up between two rooms in my house. Both rooms are carpeted, leaving flooring where the wall was. We have built up the floor in the space to match the level of the existing floors in each room. I have remnants from having the carpet installed and I am wondering how to best patch the area. Should I add padding to level off the patch with the existing carpet. Would I use two sided sticking tape or glue? Also, should the remnant be cut a little larger to blend in with existing carpet and then is there a tool to use to help it blend in? Thank you for any assistance.

Overlay the gap with a piece of remnant larger than the gap.
Temporarily tack it down .
Use a straightedge as a guide to cut through both layers of carpet with a sharp utility knife
The patch & cutout will be identical.
Same method as used to inset contrasting carpet designs
Use padding to maintain thickness.
Seaming tape should be available at home center.

If your rooms have padding you will definitely want to use padding
to make it even. They do make carpet glue. You might have to use those tack strips like they put around the edges of carpeting.
Cut your lines as straight as possible, better to cut a little larger
until you get piece in there. This is not real easy to do to make it
look perfect. Make sure to use carpet knife. good luck.