I am given a bunch of charges and asked for the electric field at a field point far away (see image below).

What I would do is the following:

What I have done is applying Gauss' Law. However, in the lecture, another approach was used.

Firstly, the charges were treated per pairs , as dipoles. And as there are 4 +ive and 4 -ive, they cancel each other out and there is no first order contribution (i.e. $\frac{1}{r}$) .

Secondly, it was argued that this a quadrupole (using the same logic that above). Thus, the final answer would qualitatively be:

$$E \propto \frac{1}{r^3}$$

Thus using two different methods one gets different answers; using method 1 one gets $E \propto \frac{1}{r^2}$ and using method 2 one gets $E \propto \frac{1}{r^3}$

Who is wrong and why?

Thanks.