How to factor $x^6+x^5-3\,x^4+2\,x^3+3\,x^2+x-1$ by hand? I know that 
$x^6+x^5-3\,x^4+2\,x^3+3\,x^2+x-1 = (x^4-x^3+x+1)(x^2+2x-1)$ 
but I would not know how to do that factoring without a software. 
Some idea? Thank you!

Note that $(x - 1/x)^{2k}$ gives $x^{2k} + x^{-2k}$ and other symmetric terms, and $(x - 1/x)^{2 k + 1} = x^{2 k + 1} - x^{- 2 k - 1}$ and other terms, it looks like due to the symmetry of the coefficients by dividing by $x^3$ you can reduce the degree to a cubic in $x - 1/x$, and go from there.
