jmc

The store's revenue is given by: number of books sold $\times$ price of each book, or $p(128-4p)=128p-4p^2$. We want to maximize this expression by completing the square. We can factor out a $-4$ to get $-4(p^2-32p)$.

To complete the square, we add $(32/2{)}^2=256$ inside the parentheses and subtract $-4\cdot 256=-1024$ outside. We are left with the expression $$-4(p^2-32p+256)+1024=-4(p-16{)}^2+1024.$$Note that the $-4(p-16{)}^2$ term will always be nonpositive since the perfect square is always nonnegative. Thus, the revenue is maximized when $-4(p-16{)}^2$ equals 0, which is when $p=16$. Thus, the store should charge $\boxed{16}$ dollars for the book.

Alternatively, since the roots of $p(128-4p)$ are 0 and 32, symmetry tells us that the extreme value will be at $p=16$. Since the coefficient on $p^2$ is negative, this is a maximum.