<s>[INST] Factorize : $x^6 − 10x^3 + 27$ I want to factorize $$x^6 − 10x^3 + 27$$ 
I tried two methods , first I let $y=x^3 $ and converted it into a quadratic but the solutions are not real . 
The second method I tried was getting it to the form $a^3+b^3+c^3-3abc$ but I could only get close ( $(x^2)^3+(x)^3+3^3-3(x^2)(x)(3)$ but that is actually $x^6-8 x^3+27$ , close but not exact )  
I'm run out of ideas , so any help is appreciated .
Note : I know that the answer is $$(x^2+2 x+3) (x^4-2 x^3+x^2-6 x+9)$$ , but I want to know how !
 [/INST] Seems to me in your attempted form $a^3+b^3 + c^3 - 3abc$ intuitively you came quite close, but quit too soon.
$x^6 − 10x^3 + 27 = (x^2)^3 + (2x)^3 +(3)^3 - 3\cdot (x^2) (2x)(3)  $
The rest I presume you can factorise.  
Of course such tricks work only on a few polynomials like the one in your question(s).
</s>