A = \(\left(1-\dfrac{4}{a^2}\right)\left(1-\dfrac{4}{b^2}\right)\)

= \(\dfrac{\left(a-2\right)\left(a+2\right)\left(b-2\right)\left(b+2\right)}{a^2b^2}\)

= \(\dfrac{ab\left(a+2\right)\left(b+2\right)}{a^2b^2}\)

= \(\dfrac{ab+2\left(a+b\right)+4}{ab}\)

= \(1+\dfrac{8}{ab}\)

\(\le1+\dfrac{8}{\left(\dfrac{a+b}{2}\right)^2}=1+8=9\)