\dfrac{2a+2}{a-2}÷(a-1)+\dfrac{a^2-1}{a^2-2a+1}

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

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

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

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

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