<s>[INST] How to factorized this 4th degree polynomial? I need your help to this polynomial's factorization.
Factorize this polynomials which doesn't have roots in Q. 
$ \ f(x) = x^4 +2x^3-8x^2-6x-1 $ 
P.S.) Are there any generalized method finding 4th degree polynomials factor?
 [/INST] Just working through the hint in the accepted answer, and verified with the Wolfram result in the second answer. Just for me, and maybe a check/guide for someone else:

!$x^4+2x^3+x^2−9x^2−6x−1 \\
(x^2+x)^2  - (3x+1)^2 \\
\\
((x^2+x) + (3x+1)) ((x^2+x)  - (3x+1)) \\
(x^2+4x+1) (x^2-2x-1) \\
\\
\text{quadratic formula for } (x^2+4x+1)\\
-4/2\pm\sqrt{16-4}/2 \\
-2 \pm\sqrt3 \\
\\
\text{quadratic formula for } (x^2-2x-1) \\
--2/2\pm\sqrt{4-4(-1)}/2 \\
1\pm\sqrt{4+4}/2 \\
1\pm\sqrt2 \\
\\
\text{factored form }\\
(x+2+\sqrt3)(x+2-\sqrt3)(x-1+\sqrt2)(x-1-\sqrt2)\\
$

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