$$x^4+2x^3-x^2-2x+1 = 0$$
Answer
$$ \begin{matrix}x_1 = 0.61803 & x_2 = -1.61803 \\[1 em] \end{matrix} $$
Explanation
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using quartic formulas
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Polynomial Equations Solver