$$x^4+10x^3+37x^2+78x+150 = 0$$
Answer
$$ \begin{matrix}x_1 = -0.39859+2.51144i & x_2 = -0.39859-2.51144i & x_3 = -4.60141+1.42285i \\[1 em] x_4 = -4.60141-1.42285i & \\[1 em] \end{matrix} $$
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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