Question
$$4x+\frac{4}{x}+1 = \frac{4}{x}$$
Answer
$$ \begin{matrix}x_1 = 0 & x_2 = -\dfrac{ 1 }{ 4 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} 4x+\frac{4}{x}+1 &= \frac{4}{x}&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]x\cdot4x+x\cdot\frac{4}{x}+x\cdot1 &= x\cdot\frac{4}{x}&& \text{cancel out the denominators} \\[1 em]4x^2+4+x &= 4&& \text{simplify left side} \\[1 em]4x^2+x+4 &= 4&& \text{move all terms to the left hand side } \\[1 em]4x^2+x+4-4 &= 0&& \text{simplify left side} \\[1 em]4x^2+x+4-4 &= 0&& \\[1 em]4x^2+x &= 0&& \\[1 em] \end{aligned} $$
In order to solve $ \color{blue}{ 4x^{2}+x = 0 } $, first we need to factor our $ x $.
$$ 4x^{2}+x = x \left( 4x+1 \right) $$$ x = 0 $ is a root of multiplicity $ 1 $.
The second root can be found by solving equation $ 4x+1 = 0$.
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