First of all, I love the box. I use it to multiply (I ban the word "foil" in my room) and factor, I briefly talk about it with completing the square - so why not division. I first saw the possibility on this post. I about lost my mind with excitement. Last semester I tried it for the first time and loved it.
Dividing by a linear factor? Use the box
Dividing by a linear factor w/ a coefficient other than 1? Use the box
Dividing by a quadratic/cubic/anything?? Use the box!
It always works. No tricks to remember. No extra steps. No extra multiplying/dividing. No times when you have to do something different.
The easiest way to understand is to look at the link above. I've also included the video of me explaining it to my students for their notes. I've also linked to the guided notes they take.
Students have also had great success with this and have found it easy. They've even taught their friends in other classes how to use this method =). Success!
Enjoy! Spread the word!
PDF of notes