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

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