In the presence of peroxide, styrene reacts with HBr to give X . When X reacted with magnesium in dry ether…

First step of the reaction involves free radical addition of HBr to styrene in presence of peroxide to form 1-bromo-2-phenyl ethane(X)

1-bromo-2-phenyl ethane on reaction with Mg in presence of dry ether forms a grignard reagent,Which reacts with CO₂ by nucleophilic addition mechansim followed by hydrolysis to form 3-phenyl propanoic acid(Y).

3-phenyl propanoic acid(Y) reacts with PCl₅ to form 3-phenyl propanoyl chloride.

3-phenyl propanoyl chloride undergoes rosenmund reduction in presence of $\mathrm{Pd}-{\mathrm{BaSO}}_4$ to form 3-phenyl propan-1-al.

Hence Z is 3-phenyl propan-1-al