What is the conceptual reasoning behind order of operations for polynomials? When you expand $(x^2 - 1)^2$ out to $(x^2)^2 - 2x^2(1)+1^2$ (i.e. $a^2 -2ab + b^2$), how do we determine whether to multiply $ab$ by $-2$, or to multiply $2ab$ together and then minus that from $(x^2)^2$?
In other words, is the equation saying "$(x^2)^2$ minus $2x^2(1) + 1^2$", or is it saying "$(x^2)^2$ with $-2x^2(1) + 1^2$"? How do we know one over the other would be the case?
Thank you~
 A: If I understand correctly, you are asking about this
$$
a^2-2ab+b^2
$$
and whether there is a 'plus $-2ab$' or 'minus $2ab$'. The short answer is that it doesn't matter—they mean exactly the same thing. This is because subtraction is addition of a negative number. It is absolutely fine to think of subtraction as taking away, but you must remember that $a-2ab$ is just a shorthand for $a+(-2ab)$. Here is an excerpt from the Wikipedia article on subtraction:

In advanced algebra and in computer algebra, an expression involving subtraction like A − B is generally treated as a shorthand notation for the addition A + (−B). Thus, A − B contains two terms, namely A and −B. This allows an easier use of associativity and commutativity.

All that being said, I think it is more intuitive to think of subtraction as taking away, and so I would generally think of $a^2-2ab$ as taking $2ab$ away from $a^2$. But it would be equally valid to think of this as addition of $-(2ab)$. Indeed, one could argue it is more valid, given that it is how subtraction is formally defined.
To answer another of your questions:

How do we determine whether to multiply $ab$ by $−2$, or to multiply $2ab$ together and then minus that from $(x^2)^2$?

Again, it doesn't matter, and so nothing really needs to be determined. Let's say $a=7$ and $b=3$. Here are two equally valid methods of interpreting $a^2-2ab$:
Method $1$ (Subtraction is taking away):
$$
a^2-2ab=a^2-2(7)(3)=a^2-42
$$
Method $2$ (Subtraction is adding a negative number):
$$
a^2-2ab=a^2+(-2ab)=a^2+(-2\times7\times3)=a^2+(-42)=a^2-42
$$
(Note that the final step, where we rewrite $a^2+(-42)$ as $a^2-42$ is perhaps more of an aesthetic choice than anything else. $a^2-42$ can seen as a neat shorthand for $a^2+(-42)$.)
In method $2$, we multiply $ab$ by $-2$, and then perform the subtraction. But because these methods are equally valid, we get the same final answer.
