Use the Lagrange multiplier method to find the maximum possible volume for a rectangular box inscribed in the ellipsoid $2x^2 + 3y^2+4z^2=12$

The formula you have for the volume ($V=xyz$) is the volume of the portion of the box in the first octant. But there are 8 octants, each symmetric to the first, so the volume should really be calculated as

$$V=8xyz$$

Updating your solution with the new objective function, it is clear that $x,y,z$ do not change--only the lagrange multiplier $\lambda$ will be 8 times its previous value.

Plugging in the $x,y,z$ you found, we get that the volume is $\frac{16\sqrt{6}}{3}$, as the given solutions indicate.

To help with the intuition surrounding this problem, we can reduce it to the 2D case. Consider the (poorly drawn) image below, and let $(x,y)$ be the coordinates of the upper right corner of the red box.

Then you have that the area of the red box is $A=xy$, but the acutal area of the insctibed rectangle is the orange box, given by $A=4xy$, doubling once for each dimension.

In the 3D case, there are 3 dimensions in which the volume needs to be doubled, but the concept is the same.