This can be written: (1/w-1/v)/(2/w^2 + 1/v).
Multiply top and bottom by vw^2: (vw-w^2)/(2v+w^2)=w(v-w)/(2v+w^2).
So you multiplied by 2vw^2 instead of vw^2, and you had 2w^-2 as 1/2w^2 instead of 2/w^2. There is a plus sign in the denominator instead of the minus in your book, which means the book's answer is wrong or the question should have had minus in the denominator instead of plus.