Step 1: Reaction of styrene with \(HBr\) in presence of peroxide.
Styrene is
\[
C_6H_5CH=CH_2
\]
In the presence of peroxide, \(HBr\) adds by anti-Markovnikov rule.
Therefore, bromine attaches to the terminal carbon.
\[
C_6H_5CH=CH_2 \xrightarrow{HBr/ROOR} C_6H_5CH_2CH_2Br
\]
So,
\[
X=C_6H_5CH_2CH_2Br
\]
Step 2: Formation of Grignard reagent.
When \(X\) reacts with magnesium in dry ether, it forms a Grignard reagent.
\[
C_6H_5CH_2CH_2Br \xrightarrow{Mg/dry\ ether} C_6H_5CH_2CH_2MgBr
\]
Step 3: Reaction with \(CO_2\) followed by hydrolysis.
Grignard reagent reacts with \(CO_2\), followed by hydrolysis, to form a carboxylic acid with one extra carbon atom.
\[
C_6H_5CH_2CH_2MgBr \xrightarrow{CO_2/H_3O^+} C_6H_5CH_2CH_2COOH
\]
So,
\[
Y=C_6H_5CH_2CH_2COOH
\]
Step 4: Reaction of \(Y\) with \(PCl_5\).
Carboxylic acid reacts with \(PCl_5\) to form acid chloride.
\[
C_6H_5CH_2CH_2COOH \xrightarrow{PCl_5} C_6H_5CH_2CH_2COCl
\]
Step 5: Rosenmund reduction.
Acid chloride on treatment with \(H_2/Pd-BaSO_4\) undergoes Rosenmund reduction to form aldehyde.
\[
C_6H_5CH_2CH_2COCl \xrightarrow{H_2/Pd-BaSO_4} C_6H_5CH_2CH_2CHO
\]
So,
\[
Z=C_6H_5CH_2CH_2CHO
\]
Step 6: Final conclusion.
Hence, the product \(Z\) is
\[
\boxed{C_6H_5CH_2CH_2CHO}
\]