Question
$$(x+5)^4-(x-1)^4 = 80$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} (x+5)^4-(x-1)^4 &= 80&& \text{simplify left side} \\[1 em]x^4+20x^3+150x^2+500x+625-(x^4-4x^3+6x^2-4x+1) &= 80&& \\[1 em]x^4+20x^3+150x^2+500x+625-x^4+4x^3-6x^2+4x-1 &= 80&& \\[1 em]x^4+20x^3+150x^2+500x+625-x^4+4x^3-6x^2+4x-1 &= 80&& \\[1 em]24x^3+144x^2+504x+624 &= 80&& \text{move all terms to the left hand side } \\[1 em]24x^3+144x^2+504x+624-80 &= 0&& \text{simplify left side} \\[1 em]24x^3+144x^2+504x+544 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using qubic formulas.
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Equations Solver