Hint
The weighted average giving the spherical core-shell form factor is
$$
F_\mathrm{core-shell}(q,r,R,\Delta\rho_c,\Delta\rho_s) = \frac{\Delta\rho_c V(r) F_\mathrm{sphere}(q,r) + \Delta\rho_s V(R) F_\mathrm{sphere}(q,R) - \Delta\rho_s V(r) F_\mathrm{sphere}(q,r)}{\Delta\rho_c V(r) + \Delta\rho_s V(R) - \Delta\rho_s V(r)}
$$
If you take the absolute square of this, you get the form factor:
$$
P_\mathrm{core-shell}(q) = |F_\mathrm{core-shell}(q)|^2
$$