The example I made of being rigorous was excessively verbose because I was trying to demonstrate rigor to a general audience without making a lot of assumptions about background information. Mathematicians communicating with other mathematician can rely on a common knowledge of theorems and notations to write more concisely. They can factor a quadratic expression in one easily verifiable step. They do use "division", though they can drop down a layer of abstraction and explain it in terms of multiplicative inverses, and either way, they know they have to make sure the divisor is not zero.