From your total area equation, isolate $h$ so that $h$ is written in terms of $x$. Make sure you divide by the correct factor (it should include both a 4 and an $x$).

Use $V = x^2 h$. Substitute your expression for $h$ into this formula and simplify the result. Then look for the answer choice whose formula matches your simplified expression.

In Desmos, enter the relation for the material constraint and solve for $h$. For example, start from $x^2 + 4xh = 1200$ on paper, then in Desmos type an expression like h = (1200 - x^2)/(4x) so that $h$ is defined in terms of $x$.

In a new line, define the geometric volume using this height, for example: V_actual(x) = x^2 * h. Desmos will now compute the true volume for any $x$ you choose (use the slider feature for $x$).

Enter four more functions corresponding to the choices, such as VA(x) = x^2*(1200 - x)/4, VB(x) = 4x*(1200 - x^2), VC(x) = (1200 - x^2)^2/(4x), and VD(x) = x*(1200 - x^2)/4. For several $x$ values, compare each $V_\text{choice}(x)$ to V_actual(x) in a table or by overlaying graphs; the choice whose values always match V_actual(x) is the correct formula.

The box has a square base of side length $x$ and height $h$ (unknown).

• Base area = $x^2$.
• Volume of the box is

Right now, $V$ is in terms of $x$ and $h$. We need $V$ in terms of $x$ only, so we must express $h$ in terms of $x$.

The $1{,}200$ cm$^2$ of material is used for:

• the square base (area $x^2$), and
• the four side faces.

Each side face is a rectangle of width $x$ and height $h$, so its area is $xh$. There are 4 sides, so total side area is $4xh$.

So the total material area is

Solve this for $h$:

Now we have $h$ written in terms of $x$.

Substitute $h = \dfrac{1{,}200 - x^2}{4x}$ into $V = x^2 h$:

Simplify by canceling an $x$ factor from $x^2$ and the $x$ in the denominator:

This matches choice D, so the correct function is $V(x) = x(1{,}200 - x^2)/4$.