Concatenated pseudorandom generators
The idea is to use a hybrid argument. Let us first write a formula for the new PRG:
$$
G(x) = G_1(G_{3,1}(x)) G_2(G_{3,2}(x)),
$$
where $G_3(x) = G_{3,1}(x) G_{3,2}(x)$.

Suppose that we could distinguish between $G(U_k)$ and $U_{4k}$, where $U_n$ denotes $n$ random bits. Consider the following sequence:
$$
G(U_k); G_1(U_k) G_2(U_k); U_{2k} G_2(U_k); U_{4k}.
$$
By assumption, we can distinguish between the first distribution and the last distribution. Hence we can distinguish an adjacent pair of distributions.


If we can distinguish $G(U_k)$ and $G_1(U_k) G_2(U_k)$, then this gives a distinguisher for $G_3$.
If we can distinguish $G_1(U_k) G_2(U_k)$ and $U_{2k} G_2(U_k)$, then this gives a distinguisher for $G_1$.
If we can distinguish $U_{2k} G_2(U_k)$ and $U_{4k}$, then this gives a distinguisher for $G_2$.


By assumption, $G_1,G_2,G_3$ are all secure, so we obtain a contradiction.