Vtyjkyuk

For example, we have a closed box with a square bottom that is three times high as it is wide.

I have $A = 2(lw) cdot 2(l3h) cdot 2(w3h)$ as my equation

How can I isolate the width?

You have a box with a square bottom let's assume that the side of the square is $a$

Thus the height of the box is $3a$ as given in the question.

Total surface area=area of 4 walls + bottom and roof area of the square box=$4cdot (3acdot a)+2cdot (acdot a)$
$$S=12a^2+2a^2=14a^2$$

$$a=sqrtfracS14$$

You may have trouble communicating your problem due to unusual use of terminology. "Isolate" was unclear, but also I don't think there's a standard interpretation of what it means to "express an equation in terms of" a variable.

You can desire to express some quantity in terms of a variable; for example, you can try to express the surface area of the box in terms of the width.
The desired result then would be an equation with the surface area $A$ on one side of the equals sign and some expression in which $w$ appears (possibly more than one appearance of $w$) on the other side.

I would not express anything "in terms of $w$" by writing an equation that begins with "$w = cdots$".

Now if we can agree on what the words mean--preferably the same thing most mathematicians would think they mean--we can look at the problem.
You have an equation that I think was supposed to be
$$
A = 2(lw) + 2(l3h) + 2(w3h).
$$
This could be the surface area $A$ written in terms of variables named $l,$ $w,$ and $h.$ Whether this is actually the surface area depends on what $l,$ $w,$ and $h$ represent. If they are meant to be length, width, and height then the equation is wrong; just compare it to the equation for a box of arbitrary length, width, and height and you will see that when you multiply length times height (for example) you are not supposed to multiply by an extra factor of $3.$

The equation is perfectly OK, however, if the length is $l$, the width is $w$,
and the height is $3h.$

Now ask yourself why you would write $3h$ for the height. Is it because the height is $3$ times the width? What is another way to write three times the width? How about $3w$? So whatever $h$ represents, it seems it must be equal to $w.$
We can write $h$ in terms of $w$ like this: $h = w.$

Now that we have determined that $h = w,$ we can take each $h$ in the equation and replace it with the thing we just set equal to $h$, that is, $w.$ Here's the result:
$$
A = 2(lw) + 2(l3w) + 2(w3w).
$$
I would simplify this as follows:
$$
A = 2lw + 6lw + 6w^2.
$$

Now consider the statement that the bottom of the box is a square.
Presumably the length and width of the box are the lengths of the sides of its bottom (since the other dimension is height).
So that should tell you how to write the length $l$ in terms of the width $w.$
And then you can plug that into the equation, and have $A$ on the left and an expression with only constants and copies of $w$ on the right.
That's expressing the area in terms of $w.$

This does seem like a lot of work for a problem like this.
Another approach, instead of using lots of variables, is start with just one variable that you're sure you want (say, $w$ for width) and try to use it to describe as much as you can.
Draw a picture of the box and start labeling the lengths of the edges:
$w$ for the width, of course--there are four possible edges of the box that measure its width, so all four could be labeled $w$ if you want--but since at least one of those edges is an edge of the square bottom, you can label that edge $w$ and the adjacent edge of the square also $w.$
The height is three times the width, that means the height is $3w,$ use that to label an appropriate edge.
Now write the formula for the surface area of a box with those edge dimensions.
That is the approach taken in the other answer, and as you can see it leads very quickly to a result.