To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x+2$ and $2\left(x-2\right)$ is $2\left(x-2\right)\left(x+2\right)$. Multiply $\frac{1}{x+2}$ times $\frac{2\left(x-2\right)}{2\left(x-2\right)}$. Multiply $\frac{2}{2\left(x-2\right)}$ times $\frac{x+2}{x+2}$.
Since $\frac{2\left(x-2\right)}{2\left(x-2\right)\left(x+2\right)}$ and $\frac{2\left(x+2\right)}{2\left(x-2\right)\left(x+2\right)}$ have the same denominator, add them by adding their numerators.
Since $\frac{2x}{\left(x-2\right)\left(x+2\right)}$ and $\frac{4}{\left(x-2\right)\left(x+2\right)}$ have the same denominator, subtract them by subtracting their numerators.
$$\frac{2x-4}{\left(x-2\right)\left(x+2\right)}$$
Factor the expressions that are not already factored in $\frac{2x-4}{\left(x-2\right)\left(x+2\right)}$.
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x+2$ and $2\left(x-2\right)$ is $2\left(x-2\right)\left(x+2\right)$. Multiply $\frac{1}{x+2}$ times $\frac{2\left(x-2\right)}{2\left(x-2\right)}$. Multiply $\frac{2}{2\left(x-2\right)}$ times $\frac{x+2}{x+2}$.
Since $\frac{2\left(x-2\right)}{2\left(x-2\right)\left(x+2\right)}$ and $\frac{2\left(x+2\right)}{2\left(x-2\right)\left(x+2\right)}$ have the same denominator, add them by adding their numerators.
Since $\frac{2x}{\left(x-2\right)\left(x+2\right)}$ and $\frac{4}{\left(x-2\right)\left(x+2\right)}$ have the same denominator, subtract them by subtracting their numerators.
$$\frac{2x-4}{\left(x-2\right)\left(x+2\right)}$$
Factor the expressions that are not already factored in $\frac{2x-4}{\left(x-2\right)\left(x+2\right)}$.
