operational amplifier - Ideal Op-amp Question with complicated feedback resistance

The question has been asked before here: Help with ideal op-amp problem

I don't understand why we simply can't reduce $R_2$, $R_3$ and $R_4$ into one single feedback resistance:

$$R_f=R_4+R_2 \parallel R_3$$ , and then just calculate the gain: $$\frac{V_2}{V_1}=\frac{-R_f}{R_1}=-\frac{R_4+R_2||R_3}{R_1}$$

The problem is shown here:

I understand the "correct" solution and how to get to it, but don't know why my solution would be "wrong".