$$x^5+4x^4+4x^3+5x^2+2x+2 = 0$$
Answer
$$ \begin{matrix}x_1 = -3.19582 & x_2 = 0.09791+0.785i & x_3 = 0.09791-0.785i \\[1 em] x_4 = -0.5+0.86603i & x_5 = -0.5-0.86603i \\[1 em] \end{matrix} $$
Explanation
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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