<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] Note that $$(x^2 + x + 1)^2 = x^4 + 2x^3 + 3x^2 + 2x + 1$$which means that your polynomial is equal to
$$
(x^2 + x + 1)^2 - 4 \\
= (x^2 + x + 1)^2 - 2^2\\
 = (x^2 + x + 1 - 2)(x^2 + x + 1 + 2)
$$
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