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Hint

To built up the spherical core-shell formfactor, you need to use the form factor amplitude of a sphere with radius $r$ as you subunit: $$ F_{sph}(q,r) = 3\frac{\sin(qr)-qr\cos(qr)}{(qr)^3} $$ If you take the absolute square of this, you get the form factor: $$ P(q,r) = |F_{sph}(q,r)|^2 $$ For later steps, you will also need the volume of a sphere with radius $r$: $$ V_{sph}(r) = 4\pi r^3/3 $$
A note on normalization. Formfactors are not consistently normalized in textbooks and computer programs. At SAStutorials.org, we normalize the formfactor, such that $P(q\rightarrow0)=1$. Consequently, the form factor amplitude has the same normalization. But in some textbooks, the form factor is normalized to the volume.