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\frac{4x}{\left(x-2\right)\left(x+2\right)}+\frac{2x}{x+2}
Factor x^{2}-4.
\frac{4x}{\left(x-2\right)\left(x+2\right)}+\frac{2x\left(x-2\right)}{\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 \left(x-2\right)\left(x+2\right) and x+2 is \left(x-2\right)\left(x+2\right). Multiply \frac{2x}{x+2} times \frac{x-2}{x-2}.
\frac{4x+2x\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}
Since \frac{4x}{\left(x-2\right)\left(x+2\right)} and \frac{2x\left(x-2\right)}{\left(x-2\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.
\frac{4x+2x^{2}-4x}{\left(x-2\right)\left(x+2\right)}
Do the multiplications in 4x+2x\left(x-2\right).
\frac{2x^{2}}{\left(x-2\right)\left(x+2\right)}
Combine like terms in 4x+2x^{2}-4x.
\frac{2x^{2}}{x^{2}-4}
Expand \left(x-2\right)\left(x+2\right).