Thank you for asking this question Alecia -- I love to tutor questions like this.
Rest assured that you can learn calculus.
I will describe the approach you should be learning. Please comment which part(s) confuse you. if any.
Find an equation relating the thing you are trying to optimize to the variables you are allowed to manipulate.
To find that equation in the case of problems like this one, it helps to draw a diagram.
I would draw a sheet of cardboard with the congruent (or in other words identical) squares cut out of each corner. I would draw dashed lines to represent the folding lines between the inner corners of these square cut-outs. Draw the diagram big enough to visualize the problem, but not so big you have no room left for your equations. Name the quantity you can adjust. Ideally, you can limit the number of variables to just one, or find equations that allow you to reduce these equations to just one remaining unknown variable.
In this problem, the length of each of the cutout square sides would be an excellent candidate. Call it, I don't care, maybe "x". OR if you'd rather thumb your nose to tradition, call it something else. Of course, it doesn't really matter what you call it.
The 15 and the 9 have been chosen/specified for us. By drawing the diagram, you can see that the dimensions of the bottom of the box (inside the folding lines) will be a simple product of two differences.
One of the differences will be (15-2x). I am sure you can find the other one. Especially if you drew the diagram. (If you didn't draw the diagram, please do. I am serious. Draw the diagram. It is worth the trouble. Every time!)
Then the volume of the box will of course (like all prisms) be the area of the base of the prism, multiplied by the height. The height of the open-box, after folding will be? I hope you see it will be "x", or whatever you called it. Bonus points if you can draw a sketch of the resulting box, in 3d.
You should now have a formula for the volume of the box in terms of your single-variable V(x).
You can optimize that by finding which value of x causes the function to reach a maximum. Obviously at the maximum, the derivative of any single-variable function has to be either zero or undefined. This particular problem isn't going to have to worry about undefined derivatives., since x can only be something between zero and half the smallest dimension (9 inches), inclusive (just by the definition of x it belongs to the set [0 inches, 9 inches].)
Can you take the derivative of your formula with respect to x? It might be easier to multiply it out (FOIL) if you are unsure of your product rule, but you should know the product rule by now so should be able to manage it either way.
If so, do that, and set the result equal to zero. Find the solution. By the nature of this problem it is clear that the volume will be zero when x is either zero inches or half of 9 inches, so the optimal point you found must be the maximum. But if you want to or if asked to, feel free to confirm that your optimal solution represents a maximum rather than a minimum or an inflection point by evaluating the sign of the second derivative at the optimal value of x (a Second-Derivative test), or by doing an evaluation of the sign of the first derivative within the intervals to either side of it (a First-Derivative test),
After finding this optimal value of x (and optional confirming/demonstrating it yields a maximum volume), plug it back into your V(x) to find the actual answer being asked. Also evaluate each of your differences on your diagram to make a new version of your diagram illustrating the actual dimensions of this optimal box you've found. And make a statement summarizing your conclusions!
