\(\dfrac{a+2}{2a-4}+\dfrac{-4a}{a^2-4}=\dfrac{a+2}{2\left(a-2\right)}+\dfrac{-4a}{\left(a-2\right)\left(a+2\right)}\)
\(=\dfrac{\left(a+2\right)^2-4a.2}{2.\left(a+2\right)\left(a-2\right)}\) \(=\dfrac{a^2+4a+4-8a}{2\left(a+2\right)\left(a-2\right)}=\dfrac{a^2-4a+4}{2\left(a+2\right)\left(a-2\right)}\) \(=\dfrac{\left(a-2\right)^2}{2\left(a-2\right)\left(a+2\right)}=\dfrac{a-2}{2\left(a+2\right)}\)
\(\dfrac{a+2}{2a-4}+\dfrac{-4a}{a^2-4}=\dfrac{a+2}{2\left(a-2\right)}+\dfrac{-4a}{\left(a-2\right)\left(a+2\right)}\) \(=\dfrac{\left(a+2\right)^2-4a.2}{\left(a-2\right)\left(a+2\right)}=\dfrac{a^2+4a+4-8a}{\left(a-2\right)\left(a+2\right)}\) \(=\) \(\dfrac{a^2-4a+4}{\left(a-2\right)\left(a+2\right)}=\dfrac{\left(a-2\right)^2}{\left(a-2\right)\left(a+2\right)}\)
\(=\dfrac{a-2}{a+2}\)