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.

What color carpet should I replace my old carpet with?

I am going to be using remnants because I can’t afford new carpet. My room has aquamarine blue walls and light mahogany trim. I also have a lot of light lime green picture frames on the walls.

Serious answers only please.