$$x+2+\frac{2}{x-1} = 0$$
Answer
$$ \begin{matrix}x_1 = 0 & x_2 = -1 \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x+2+\frac{2}{x-1} &= 0&& \text{multiply ALL terms by } \color{blue}{ x-1 }. \\[1 em](x-1)x+(x-1)\cdot2+(x-1)\cdot\frac{2}{x-1} &= (x-1)\cdot0&& \text{cancel out the denominators} \\[1 em]x^2-x+2x-2+2 &= 0&& \text{simplify left side} \\[1 em]x^2+x-2+2 &= 0&& \\[1 em]x^2+x-2+2 &= 0&& \\[1 em]x^2+x &= 0&& \\[1 em] \end{aligned} $$
In order to solve $ \color{blue}{ x^{2}+x = 0 } $, first we need to factor our $ x $.
$$ x^{2}+x = x \left( x+1 \right) $$$ x = 0 $ is a root of multiplicity $ 1 $.
The second root can be found by solving equation $ x+1 = 0$.
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