Step 1: Understanding the structure.
The given molecule contains two double bonds with substituents that will create chiral centers upon hydrogenation. Each C=C bond is converted into a C–C bond, producing two new stereocenters.
Step 2: Calculating stereoisomers.
Number of stereocenters formed = 2.
Maximum stereoisomers = 2$^n$ = 2$^2$ = 4.
Because the molecule has no internal plane of symmetry after hydrogenation, no meso form is possible.
Step 3: Conclusion.
Thus, the number of possible stereoisomers is 4.