\Big( \dfrac{a^2}{a-2}-\dfrac{4}{a-2} \Big) \cdotp \dfrac{1}{a^2+2a}

= \dfrac{a^2-4}{a-2} \cdotp \dfrac{1}{a(a+2)}

= \dfrac{(a+2)(a-2)}{a-2} \cdotp \dfrac{1}{a(a+2)}

=\dfrac{1}{a}