First a/b - b/a=(a^2-b^2)/ab. Now square it: (a^2-b^2)^2/(ab)^2.
Now multiply by (ab)^2/(b-a)^2. The (ab)^2 cancel out leaving (a^2-b^2)^2/(b-a)^2.
We can factorise the top: ((a-b)(a+b))^2=(a-b)^2 * (a+b)^2/(b-a)^2.
(b-a)^2 is the same as (a-b)^2, so again we can cancel out (a-b)^2 leaving (a+b)^2.