$$9n^4+10-8n^5+12n^3+9-2n^5 = 0$$
Answer
$$ \begin{matrix}n_1 = 1.77085 & n_2 = 0.47698+0.84001i & n_3 = 0.47698-0.84001i \\[1 em] n_4 = -0.91241+0.56333i & n_5 = -0.91241-0.56333i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} 9n^4+10-8n^5+12n^3+9-2n^5 &= 0&& \text{simplify left side} \\[1 em]-10n^5+9n^4+12n^3+19 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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