$$x^4+x^3+2x^2+2x+1 = 0$$
Answer
$$ \begin{matrix}x_1 = -0.62174+0.4406i & x_2 = -0.62174-0.4406i & x_3 = 0.12174+1.30662i \\[1 em] x_4 = 0.12174-1.30662i & \\[1 em] \end{matrix} $$
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