$$x^4+4x^2+3 = 0$$
Answer
$$ \begin{matrix}x_1 = 1i & x_2 = -i & x_3 = 1.73205i \\[1 em] x_4 = -1.73205i & \\[1 em] \end{matrix} $$
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