This is a problem related to Poisson Process where $\lambda = 2$.
$ E(N_3N_4) \\ = E[N_3(N_4-N_3 + N_3)] \\ = E[N_3(N_4-N_3) + N^2_3)] \\ = E[N_3 - N_0(N_4-N_3) + N^2_3)] \\ = E[N_3 - N_0(N_4-N_3)] + E(N^2_3)] \\ = E[N_3 N_1] + E(N^2_3)] \\ = E[N_3] E[ N_1] + E(N^2_3)] \\ = ... $
I haven't understood this.
Where does $N_0$ suddenly come from, and how did it accommodate itself with a minus sign beside $N_3$?
Where did $N_1$ come from?
How did $E[N_3 N_1]$ become $E[N_3]E[N_1]$?
As the Poisson process is a Lévy process. We know that its increments are independent and stationary. This means that if $N_t$ is a Poisson process with intensity $\lambda$ then $\forall t \geq s$ \begin{align*} N_t - N_s \text{ is independent of} \, N_s \\ N_t - N_s \sim \mathcal{P}(\lambda(t-s)) \end{align*} Suppose that $\lambda = 2$. Let us compute $E[N_4N_3]$ \begin{align} E[N_4N_3] &= E[N_3(N_4-N_3) + N_3^2] \\ &=E[N_3]E[N_4-N_3] + E[N_3^2] \\ &=2.3.2.(4-3) + 2.3 + (2.3)^2 \\ &= 54 \end{align} Where the second equality holds due to the independent increments.