$$\frac{4}{x}+1 = \frac{2x+2}{x}$$

$$ \begin{aligned} \frac{4}{x}+1 &= \frac{2x+2}{x}&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]x\cdot\frac{4}{x}+x\cdot1 &= x \cdot \frac{2x+2}{x}&& \text{cancel out the denominators} \\[1 em]4+x &= 2x+2&& \text{move the $ \color{blue}{ 2x } $ to the left side and $ \color{blue}{ 4 }$ to the right} \\[1 em]x-2x &= 2-4&& \text{simplify left and right hand side} \\[1 em]-x &= -2&& \text{switch signs on both sides} \\[1 em]x &= 2&& \\[1 em] \end{aligned} $$

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