$$1+\frac{8}{x} = \frac{9}{x^2}$$
Answer
$$ \begin{matrix}x_1 = 1 & x_2 = -2.28547 & x_3 = 0.64274+1.87745i \\[1 em] x_4 = 0.64274-1.87745i & \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} 1+\frac{8}{x} &= \frac{9}{x^2}&& \text{multiply ALL terms by } \color{blue}{ xx^2 }. \\[1 em]xx^2\cdot1+xx^2\cdot\frac{8}{x} &= xx^2\cdot\frac{9}{x^2}&& \text{cancel out the denominators} \\[1 em]x^3+8 &= \frac{9}{x^1}&& \text{multiply ALL terms by } \color{blue}{ x^1 }. \\[1 em]x^1\cdot1x^3+x^1\cdot8 &= x^1\cdot\frac{9}{x^1}&& \text{cancel out the denominators} \\[1 em]x^4+8x &= 9&& \text{move all terms to the left hand side } \\[1 em]x^4+8x-9 &= 0&& \\[1 em] \end{aligned} $$
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