$$27a^4-18a^2+8a-5 = 0$$
Answer
$$ \begin{matrix}a_1 = 0.7707 & a_2 = -1.05532 & a_3 = 0.14231+0.45545i \\[1 em] a_4 = 0.14231-0.45545i & \\[1 em] \end{matrix} $$
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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