Answer
See the answer below.
Work Step by Step
The slow-step is rate-determining.
N2O2 is an intermediary since it's not present in the global reaction.
For the slow step
$r=r_S=k_S[O_2][N_2O_2]$
for the equilibrium back reaction:
$r_{-F}=k_{-F}[N_2O_2]=r_F=k_F[NO]^2$
$r=k_S[O_2]r_{-F}/k_{-F}=(k_Sk_F/k_{-F})[O_2][NO_2]^2$
$r=k[O_2][NO_2]^2$
