I got the following question:

Let $\{X\_n, n \in \mathbb{N} \}$ be a branching process with offspring distribution $Z$. Now let $\{N(t); t \geq 0 \}$ be a Poisson process with intensity $\lambda$. 

For fixt $s,t$ such that $0 < s < t$ assume $Z \sim N(s) +N(t)$.

Give an expression for $Var(X\_n)$.

There is a solution given

\begin{align\*} Var(Z) = Var(N(t)+N(s)) &= Var(N(t) +N(s) +N(s) -N(s)) \\
 &= Var(N(t) -N(s)) + 4Var(N(s)) \\ 
&= \lambda(t-s)+4\lambda s = \lambda(t+3s) \end{align\*}

Maybe I'm just tired but I cant wrap my head around this. I understand every stop above and why we can do it but not why it is done. 

Why is $Var(N(t)+N(s)) = Var(N(t)) +Var(N(s)) = \lambda(t+s)$ wrong?