most of what I can think of I probably already have posted somewhere like the trivial results that

edit: yep so trivial I messed up it's statement. basically I was just using the fact that:

1(2x+1)^2 = 4x^2+4x+1 = x(4x+3)+(x+1)

2(2x+1)^2 = 8x^2+8x+2 = x(8x+7)+(x+2)

which technically is:

because m=0 for the 1 case. so this is just a specific case of a more general algebraic fact.