Is there a theorem or name for the factorization $x^4 + 4 = (x^2 + 2x + 2)(x^2 - 2x + 2)$

Question

I see a common pattern where the sum of two terms to the fourth power can be reduced to two quadratic factors. Here is an example.

$ x^4 + 4 = (x^2 + 2x + 2)(x^2 - 2x + 2) $

I mean I get that it can be obtained through adding and subtracting a term so that it can become a difference of two squares and then factoring it. But I would like to know if there like a name or law that represents this concept. I think I recall seeing something to that effect in Wikipedia but right now I can't seem to find it again.

Is $4$ really a fourth product if we’re factoring over the integers? — Ted Shifrin, Dec 01 '22 at 03:41
I think you mean the sum of a fourth power and four times a fourth power — J. W. Tanner, Dec 01 '22 at 04:07
Exact Dupe of Prove $n^4+4$ is composite for all integers $n\ge 1$ or, more genera;lly, this. — Bill Dubuque, Feb 27 '23 at 22:31
It’s not an exact duplicate, because this question asked for the name — J. W. Tanner, Feb 27 '23 at 22:47

score 4 · Accepted Answer · answered Dec 01 '22 at 04:15

I would like to know if there [is] a name or law that represents this concept.

The sum of a fourth power and four times a fourth power, of which you gave an example,

can be factored using what is called Sophie Germain's identity:

$a^{4} + 4b^{4} = (a^{2}+2b^{2}-2ab)(a^{2}+2b^{2}+2ab).$

Is there a theorem or name for the factorization $x^4 + 4 = (x^2 + 2x + 2)(x^2 - 2x + 2)$

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