@user21820 Sorry I don't want to itrerupt the ongoing logic discussion, but whenever you have time: Do you know of any books which teach more advanced algebraic manipulation? I am talking about questions such as: Solve $(x^2-a)^2=x+a$, i.e. the types of problems that you always see Michael Rozenberg solve
@Ovi If it were about inequalities, there are a couple of standard techniques. But if you're asking about just algebraic manipulations, there simply aren't many general techniques, because of the intrinsic nature of algebra. For instance, have you seen the algebraic general solutions of the cubic and quartic? The first step of reducing to depressed form can be vaguely motivated by saying that we want to reduce the number of free parameters. But beyond that it is thoroughly ad-hoc.
I have once asked some people whether they could motivate these algebraic substitutions via Galois theory. They didn't give me an answer. Galois theory in fact motivates a different kind of solution via Lagrange resolvents.
I edited it to ping him. Let's see whether he can or not. I asked him about a previous one. If he can see this one then you and he clearly have some difference.
@user21820 Hmm that is a bit sad :(. But aren't inequalities just about using those same algebraic manipulations to bring your problem in the form of a known inequality?