If $2 x^4 + x^3 - 11 x^2 + x + 2 = 0$, then find the value of $x + \frac{1}{x}$? 
If $2 x^4 + x^3 - 11 x^2 + x + 2 = 0$, then find the value of $x + \frac{1}{x}$ ?

I would be very grateful if somebody show me how to factor this polynomial by hand, as of now I have used to Mathematica to get  $(x-2) (2x - 1) (1 + 3x + x^2)$,  but I am having trouble to factor this manually.
Best regards,

You don't need to factor the polynomial to find the value of $x + \frac{1}{x}$.  Note that the given condition implies $\left( 2x^2 + \frac{2}{x^2} \right) + \left( x + \frac{1}{x} \right) - 11 = 0$ by dividing by $x^2$ and collecting symmetric terms.  Now observe that $\left( x + \frac{1}{x} \right)^2 = x^2 + \frac{1}{x^2} + 2$...
But if you want to factor it by hand anyway, your best friend is the rational root test.
