$$ \begin{aligned} \frac{x}{x}+2-\frac{1}{x}-2 &= \frac{8}{x^2}-4&& \text{multiply ALL terms by } \color{blue}{ xx^2 }. \\[1 em]xx^2\frac{x}{x}+xx^2\cdot2-xx^2\cdot\frac{1}{x}-xx^2\cdot2 &= xx^2\cdot\frac{8}{x^2}-xx^2\cdot4&& \text{cancel out the denominators} \\[1 em]x+2x^3-1-2x^3 &= \frac{8}{x^1}-4x^3&& \text{multiply ALL terms by } \color{blue}{ x^1 }. \\[1 em]x^1\cdot1x+x^1\cdot2x^3-x^1\cdot1-x^1\cdot2x^3 &= x^1\cdot\frac{8}{x^1}-x^1\cdot4x^3&& \text{cancel out the denominators} \\[1 em]x^2+2x^4-x-2x^4 &= 8-4x^4&& \text{simplify left and right hand side} \\[1 em]x^2+2x^4-x-2x^4 &= -4x^4+8&& \\[1 em]x^2-x &= -4x^4+8&& \text{move all terms to the left hand side } \\[1 em]x^2-x+4x^4-8 &= 0&& \text{simplify left side} \\[1 em]4x^4+x^2-x-8 &= 0&& \\[1 em] \end{aligned} $$

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