$$x^4+10x^3+37x^2+78x+1 = 0$$
Answer
$$ \begin{matrix}x_1 = -0.0129 & x_2 = -5.99331 & x_3 = -1.99689+2.99125i \\[1 em] x_4 = -1.99689-2.99125i & \\[1 em] \end{matrix} $$
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