The "trick" for solving a binomial with a negative index is given by:
I understand how to expand binomials of the form $1+x$ for negative indices, but I do not why this method of factoring works for binomials of form $a+x$ with negative exponents.
(Note: I have not yet learned Newton's Binomial Theorem but I have learned the case for $1+a$ with negative integers, so please do not include Newton's Therem in your answer if it's possible. Also, I understand that they have divided the binomial by $a$ or $x$ in each expression but I don't understand the factors that have been taken out.)
Lately, I have noticed that if I express the binomial expression as a reciprocal with positive powers the answer maintains correct, but I'm not certain if this is valid for all negative exponents.