If you look at any list of elementary algebra topics, or any book's table of contents, etc., then you'll probably find that all of the subjects are referenced by name except for one single exceptional case, which is always expressed in symbolic form. For example, from the College Board's Accu-Placer Program Manual, here's a list of Content Areas for the Elementary Algebra test:
Do you see it? Or, here are some of the section headers in the Pearson testbank which accompanies the Martin-Gay Prealgebra & Introductory Algebra text:
Or, here's a menu of topics and quizzes from the MathGuide.com algebra site:
I could repeat this for many other cases, such as: the CUNY list of elementary algebra topics, tables of contents for most algebra books, etc., etc. It's weird and to my OCD brothers and sisters surely it's a bit distracting and frustrating.
There should be a name for this. The funny thing is that, to my current understanding, there's a perfectly serviceable name to make the distinction that we're reaching for here: "monic" means a polynomial with a lead coefficient of 1. So I've taken to, in my classes, referring to the initial or "basic" type (\(x^2 + bx + c\)) as a monic quadratic, and the more general or "advanced" type (\(ax^2 + bx + c\), \(a \ne 1\)) as a nonmonic quadratic. My students know they must learn proper names for everything, and so they pick this up as easily as anything else, and without complaint. Thereafter it's much easier to communally reference the different structures by their proper names.
Now: I must admit that I picked this up from Wikipedia and I've never, ever, seen it used in any mathematics textbook at any level. Perhaps someone could tell me if this is new, or nonstandard, or inaccurate. But even if that weren't the right term to distinguish a polynomial with lead coefficient 1, there should still be a name for this structure. We really should create a name, if necessary, and I'd be prone to make up my own name for something like that.
But "monic" fits perfectly and is delightfully short and descriptive. We should all start using "monic" more widely, and I'd love to start seeing it in major algebra textbooks.