$$\frac{1269}{2000}m^6-\frac{223}{50}m^3-\frac{129}{50}m+\frac{784}{25} = 0$$
Answer
$$ \begin{matrix}m_1 = 1.78028+0.59734i & m_2 = 1.78028-0.59734i & m_3 = -0.27558+1.87944i \\[1 em] m_4 = -0.27558-1.87944i & m_5 = -1.50471+1.27294i & m_6 = -1.50471-1.27294i \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1269}{2000}m^6-\frac{223}{50}m^3-\frac{129}{50}m+\frac{784}{25} &= 0&& \text{multiply ALL terms by } \color{blue}{ 2000 }. \\[1 em]2000 \cdot \frac{1269}{2000}m^6-2000\frac{223}{50}m^3-2000\frac{129}{50}m+2000\cdot\frac{784}{25} &= 2000\cdot0&& \text{cancel out the denominators} \\[1 em]1269m^6-8920m^3-5160m+62720 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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