Step 1: Look at acrolein.
Acrolein is $\text{CH}_2=\text{CH}-\text{CHO}$, and every one of its three carbons takes part in a double bond, either C=C or C=O, so all three carbons are sp2.
Step 2: Look at formaldehyde.
Formaldehyde, $\text{H}_2\text{C}=\text{O}$, has just one carbon, and it is sp2 because of the C=O double bond.
Step 3: Look at peroxyacetyl nitrate (PAN).
PAN is $\text{CH}_3-\text{CO}-\text{OO}-\text{NO}_2$. The methyl carbon is sp3, single bonded on all sides, while the carbonyl carbon next to it is sp2 due to its C=O bond, giving one sp2 carbon here.
Step 4: Add them all up.
$3$ (acrolein) $+ 1$ (formaldehyde) $+ 1$ (PAN) $= 5$.
\[ \boxed{5 \text{ sp}^2 \text{ carbons}} \]