$$\frac{4}{x}+1 = 2x+\frac{2}{x}$$
Answer
$$ \begin{matrix}x_1 = \dfrac{ 1 }{ 4 }-\dfrac{\sqrt{ 17 }}{ 4 } & x_2 = \dfrac{ 1 }{ 4 }+\dfrac{\sqrt{ 17 }}{ 4 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{4}{x}+1 &= 2x+\frac{2}{x}&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]x\cdot\frac{4}{x}+x\cdot1 &= x\cdot2x+x\cdot\frac{2}{x}&& \text{cancel out the denominators} \\[1 em]4+x &= 2x^2+2&& \text{move all terms to the left hand side } \\[1 em]4+x-2x^2-2 &= 0&& \text{simplify left side} \\[1 em]-2x^2+x+2 &= 0&& \\[1 em] \end{aligned} $$
$ -2x^{2}+x+2 = 0 $ is a quadratic equation.
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