The combination of NO_2(g) and CO(g) occurs in two steps as...

The rate of a complex reaction is determined by the slowest step, which is known as the rate-determining step.

In the given mechanism, the first step is the slow step:
$2\text{NO}_2 \rightarrow \text{NO} + \text{NO}_3$

For an elementary step, the rate law is written directly from its stoichiometry according to the law of mass action.

Rate $= k_1[\text{NO}_2]^2$

Answer: Rate $= k_1[\text{NO}_2]^2$