\(A=\left(\dfrac{4}{\left(x-2\right)\left(x+2\right)}+2\right)\cdot\dfrac{x+2}{2x}+\dfrac{2}{x-2}\)
\(=\dfrac{4+2x^2-8}{\left(x-2\right)\left(x+2\right)}\cdot\dfrac{x+2}{2x}+\dfrac{2}{x-2}\)
\(=\dfrac{2\left(x^2-2\right)}{\left(x-2\right)\left(x+2\right)}\cdot\dfrac{x+2}{2x}+\dfrac{2}{x-2}\)
\(=\dfrac{\left(x^2-2\right)}{x\left(x-2\right)}+\dfrac{2}{x-2}\)
\(=\dfrac{x^2+2x-2}{x\left(x-2\right)}\)
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