Question
$$y^6-1686y^5-525624y^4+71116y^3+36y^2-20-6 = 0$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} y^6-1686y^5-525624y^4+71116y^3+36y^2-20-6 &= 0&& \text{simplify left side} \\[1 em]y^6-1686y^5-525624y^4+71116y^3+36y^2-26 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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Polynomial Equations Solver