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Transforming $(a+b)(a-b)$ into $a^2-b^2$ is easy: three distributions, two associations, one cancellation, and two factorings.

Going from $a^2-b^2$ to $(a+b)(a-b)$ is harder: You have to know to invent the term $ab$, which is then both added and subtracted.

How do you call that invent something, and then both do it and undo it trick?

Do the same thing to both sides of an equation is one special case of it, but that doesn't fit here.

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    $\begingroup$ I usually call it the "add a clever form of zero" trick. $\endgroup$ Commented Sep 9 at 11:58
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    $\begingroup$ Like @StevenGubkin, I call it "Creatively Adding Zero" and its companion "Creatively Multiplying by 1". Whenever this comes up initially in a class, the first few times, I also point out that it's allowed to, say add and subtract $7a^3b^5$ if I wanted -- it is just not useful in this particular problem; our goal is to find the useful version of zero. $\endgroup$ Commented Sep 9 at 12:52
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    $\begingroup$ Why would any one go this way to show $a^2-b^2=(a+b(\cdot(a-b)$ instead of dividing $a^2-b^2 $ by $a-b$ or just using to know what $a^2-b^2$ is ,wich you learn early in high school? I know this is no answer to the question, I just wondered. $\endgroup$ Commented Sep 9 at 17:26
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    $\begingroup$ One example where this trick gets used is in proving the product rule: in the numerator of the difference quotient one rewrites $f(x+h)g(x+h) - f(x)g(x)$ as $(f(x+h)g(x+h) - f(x)g(x+h)) + (f(x)g(x+h) - f(x)g(x))\\$, then factors each grouped pair of terms and takes the limit. $\endgroup$ Commented Sep 10 at 3:40
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    $\begingroup$ I've also heard called "a convenient form of zero." $\endgroup$ Commented Sep 11 at 16:15

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Adding a zero

We know that $A = A+0$ and we know that $0 = B-B$, so we also know $A = A + B - B$. (Some stages omitted for brevity.)

I am used to calling this adding a zero (nollan lisääminen in Finnish, the languages I am used to thinking about this in), or possible adding a zero in a clever way, or some variant thereof. This is probably related to the axioms (of, say, natural or real numbers, or possibly theorems deduced from them), as it follows so straightforwardly and nicely from them.

This is kind of a cute and technically correct name. I am not sure it is good pedagogically for all students and pupils at all levels.

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