$$2+\frac{1}{n} = \frac{1}{n+2}$$

$$ \begin{aligned} 2+\frac{1}{n} &= \frac{1}{n+2}&& \text{multiply ALL terms by } \color{blue}{ n(n+2) }. \\[1 em]n(n+2)\cdot2+n(n+2)\cdot\frac{1}{n} &= n(n+2)\cdot\frac{1}{n+2}&& \text{cancel out the denominators} \\[1 em]2n^2+4n+n+2 &= n&& \text{simplify left side} \\[1 em]2n^2+5n+2 &= n&& \text{move all terms to the left hand side } \\[1 em]2n^2+5n+2-n &= 0&& \text{simplify left side} \\[1 em]2n^2+4n+2 &= 0&& \\[1 em] \end{aligned} $$

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