Question
$$\frac{1}{x}-1+\frac{1}{x}+1+2\frac{x}{x^2}+1+4\frac{x^3}{x^4}+1 = 0$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} \frac{1}{x}-1+\frac{1}{x}+1+2\frac{x}{x^2}+1+4\frac{x^3}{x^4}+1 &= 0&& \text{multiply ALL terms by } \color{blue}{ xx^2x^4 }. \\[1 em]xx^2x^4\cdot\frac{1}{x}-xx^2x^4\cdot1+xx^2x^4\cdot\frac{1}{x}+xx^2x^4\cdot1+xx^2x^4\cdot2\frac{x}{x^2}+xx^2x^4\cdot1+xx^2x^4\cdot4\frac{x^3}{x^4}+xx^2x^4\cdot1 &= xx^2x^4\cdot0&& \text{cancel out the denominators} \\[1 em]x^2-x^7+x^2+x^7+2+x^7+\frac{4}{x^4}+x^7 &= 0&& \text{multiply ALL terms by } \color{blue}{ x^4 }. \\[1 em]x^4\cdot1x^2-x^4\cdot1x^7+x^4\cdot1x^2+x^4\cdot1x^7+x^4\cdot2+x^4\cdot1x^7+x^4\cdot\frac{4}{x^4}+x^4\cdot1x^7 &= x^4\cdot0&& \text{cancel out the denominators} \\[1 em]x^6-x^{11}+x^6+x^{11}+2x^4+x^{11}+4+x^{11} &= 0&& \text{simplify left side} \\[1 em]x^6-x^{11}+x^6+x^{11}+2x^4+x^{11}+4+x^{11} &= 0&& \\[1 em]2x^{11}+2x^6+2x^4+4 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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