Start by writing an expression for work in terms of just v and p.

Expand the momentum part (careful here: both m and v are variables) by substituting an expression for the mass.

Note that work is done on both the rocket and the exhaust.  When these integrals are added together, some terms cancel and/or combine.

The solution is: work = (final mass) x (exhaust velocity) x (final rocket velocity)

This is a tough problem - feel free to help each other and/or see me for further assistance!