<s>[INST] Factorization of quartic polynomial. I want to know other ways of factorization to get quadratic factors in this polynomial:
$$x^4+2x^3+3x^2+2x-3$$
Thanks in advance for your suggestions. 
The original polynomial is $$x^6-2x^3-4x^2+8x-3$$ where the two found factors are $(x+1)$ and $(x-1)$ by synthetic division. 
 [/INST] Splitting up the x^2 term works well for factorising quartic but one has to be smart in grouping terms
In this , write 3x^2 as x^2 + 2x^2 and group terms as 
=  x^2(x^2+2x+1) +2x^2+2x-3  =[x(x+1)]^2 +2x(x+1) -3
= [x(x+1) +3][x(x+1)-1]
</s>