Solving a 6th degree polynomial equation I have a polynomial equation that arose from a problem I was solving. The equation is as follows:

$$-x^6+x^5+2x^4-2x^3+x^2+2x-1=0 .$$

I need to find $x$, and specifically there should be a real value where $\sqrt3<x<\sqrt{2+\sqrt2}$, in accordance to the problem I am solving. I know that it would be possible for me to find approximations of the roots of the equation, but I would prefer to know the exact value of this specific root (i.e. with the answer as a surd, with nested surds if required). I am unable to do this as I do not know any method of solving polynomials of degree $> 4$.
If this cannot be done, could you tell me an approximate decimal value of $x$, or at least check that a solution exists within the range I have given (it is possible that I made an error earlier in my algebra).

Using a complicated computer algebra system (Mathematica 10.4), we can get all the roots to this equation as radicals.  Two roots are complex and the rest are all real (surprisingly).
Module[{roots},
  roots = Solve[-x^6 + x^5 + 2 x^4 - 2 x^3 + x^2 + 2 x - 1 == 0, x];
  Transpose[{
    N[x /. roots],
    FullSimplify[Element[x, Reals] /. roots],
    x /. roots
  }]
] // TableForm



*

*There are two complex roots at $\frac{1}{2} \pm \mathrm{i}\sqrt{3}$.

*There is a real root at $-1$.

*The other three are complicated and real: \begin{align}
1.80194\dots{} &= \frac{1}{3} \left(1+\frac{7^{2/3}}{\sqrt[3]{\frac{1}{2} \left(-1+3 \mathrm{i}
   \sqrt{3}\right)}}+\sqrt[3]{\frac{7}{2} \left(-1+3 \mathrm{i}
   \sqrt{3}\right)}\right),  \\
   -1.24698\dots{} &= \frac{1}{3}-\frac{7^{2/3} \left(1+\mathrm{i}
   \sqrt{3}\right)}{3\ 2^{2/3} \sqrt[3]{-1+3 \mathrm{i} \sqrt{3}}}-\frac{1}{6}
   \left(1-\mathrm{i} \sqrt{3}\right) \sqrt[3]{\frac{7}{2} \left(-1+3 \mathrm{i}
   \sqrt{3}\right)},  \\
   0.445042\dots{} &= \frac{1}{3}-\frac{7^{2/3} \left(1-\mathrm{i} \sqrt{3}\right)}{3\
   2^{2/3} \sqrt[3]{-1+3 \mathrm{i} \sqrt{3}}}-\frac{1}{6} \left(1+\mathrm{i} \sqrt{3}\right)
   \sqrt[3]{\frac{7}{2} \left(-1+3 \mathrm{i} \sqrt{3}\right)}  \text{.}
\end{align}


Conveniently, the first one is in the interval you require.
These three complicated roots are the roots of the same polynomial @Travis gets.
Looking at the Galois group structure, I believe we cannot dispense with complex numbers in these expressions.
