Question
$$x^6+4x^5+8x^4+1+4x+8x^2 = 140x^3$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} x^6+4x^5+8x^4+1+4x+8x^2 &= 140x^3&& \text{simplify left side} \\[1 em]x^6+4x^5+8x^4+8x^2+4x+1 &= 140x^3&& \text{move all terms to the left hand side } \\[1 em]x^6+4x^5+8x^4+8x^2+4x+1-140x^3 &= 0&& \text{simplify left side} \\[1 em]x^6+4x^5+8x^4-140x^3+8x^2+4x+1 &= 0&& \\[1 em] \end{aligned} $$
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