<s>[INST] How can we describe this factorization process? We know that, there exist infinite number of algebraic expressions, so that we can not factorise these with real coefficients. For example,

*

*$a^2+b^2$


*$a^2+b^2+c^2$
The simplest example, $a^2+b^2$ can not be factored over $\Bbb R$.
But, I want to consider the factorization like the one below.
$$a^2+b^2=(a+b)^2-2ab=(a+b-\sqrt {2ab})(a+b+\sqrt{2ab})$$
But, what is the mathematical name of this type of factorisation? Is this a really factoring operation?
Edit:
Ah, I see that the, people misunderstood my question, maybe I was not clear.
I know that,
$$a^2+b^2=(a+b)^2-2ab=(a+b)^2-(\sqrt {2ab})^2=(a+b-\sqrt {2ab})(a+b+\sqrt{2ab})$$
I am aware $$a^2-b^2=(a-b)(a+b)$$
But, my question is different here.
I wonder how this factoring operation is positioned in mathematics.
I mean, in which mathematical sense, this can be considered as factoring?
$$a^2+b^2=(a+b-\sqrt {2ab})(a+b+\sqrt{2ab})$$
How can we describe/define this factorization process?
$$a^2+b^2=(a+b-\sqrt {2ab})(a+b+\sqrt{2ab})$$ over $?$
 [/INST] Note that if
$a=c^2$ and $b=2d^2$
then
$a^2+b^2=(a+b)^2-2ab=(a+b-\sqrt {2ab})(a+b+\sqrt{2ab})$
becomes
$c^4+4d^4
=(c^2+2d^2)^2-4c^2d^2
=(c^2+2d^2-2cd)(c^2+2d^2+2cd)
$
which is an
unexpected factorization
with integer coefficients.
</s>