If the base is a square, the length is equal the width (let's call it x).
If the height is 2 ft greater than the length and width, we have:

Also, if the surface area is 384 ft², we have:

Using h = x + 2 in this equation, we have:

Solving this equation using the quadratic formula, we have:
![\begin{gathered} x_1=\frac{-b+\sqrt[]{b^2-4ac}}{2a}=\frac{-8+\sqrt[]{64+7680}}{10}=(-8+88)/(10)=8 \\ x_2=(-8-88)/(10)=-9.6 \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/high-school/uvnziqccsd2rx57jz1jr5r3zipoejgwc5v.png)
So if x = 8, the dimensions of the box are 8 ft of length, 8 ft of width and 10 ft of height.