<s>[INST] Factoring a quartic polynomial over the integers with roots that are not integers The quartic polynomial
$$
1728(x - 3) - x^2(12^2 - x^2)
$$
factors "nicely" as
$$
(x^2 - 12x + 72) (x^2 + 12x - 72) = (x^2 - 12x + 72)(x - 6\sqrt{3} + 6)(x + 6\sqrt{3} + 6) \, .
$$
(Note that $1728 = 3(24^2)$.)  How is this factorization obtained?
 [/INST] We have
$$
(x^2-12x+72)(x^2+12x-72)=x^4 - 144x^2 + 1728x - 5184=x^2(x^2-12^2)+1728(x-3),
$$
The factorization is of the form $(a^2+b)(a^2-b)=a^4-b^2$, with $a=x$ and $b=12x-72$.
</s>