Let's work through the problem step by step and resolve the error.

Given:

• A rectangular piece of cardboard measuring 12 inches by 18 inches.
• Squares with side length x inches are cut out from each corner.

Part a:

1. Width of the box:
   The width is reduced by twice the square's side length (since squares are cut from both sides):
   $\text{Width} = 12 - 2x \, \text{inches}$

2. Height of the box:
   The height of the box is simply x since the sides bent upward form the height: $\text{Height} = x \, \text{inches}$

Part b:

3. Length of the box:
   The length is also reduced by twice the square's side length: $\text{Length} = 18 - 2x \, \text{inches}$

4. Volume of the box:
   The volume $V$ of a rectangular box is given by: $V = \text{Length} \times \text{Width} \times \text{Height}$ Substituting the expressions for length, width, and height: $V = (18 - 2x)(12 - 2x)(x)$ Let's expand this expression:

   First, expand $(18 - 2x)(12 - 2x)$: $(18 - 2x)(12 - 2x) = 216 - 36x - 24x + 4x^2 = 216 - 60x + 4x^2$ Now, multiply the result by $x$: $V = x(216 - 60x + 4x^2) = 4x^3 - 60x^2 + 216x$ Thus, the volume is: $V = 4x^3 - 60x^2 + 216x \, \text{cubic inches}$

Part c:

5. Outside surface area of the box (no top):
   The surface area includes the bottom and the four sides. The bottom has dimensions Length × Width, and each side has an area of Height × respective side length.

   • Bottom area: $\text{Bottom area} = (18 - 2x)(12 - 2x)$ We already expanded this: $\text{Bottom area} = 216 - 60x + 4x^2$

   • Side areas:

     • Two sides have an area of $\text{Height} \times \text{Width} = x(12 - 2x) = 12x - 2x^2$
     • Two other sides have an area of $\text{Height} \times \text{Length} = x(18 - 2x) = 18x - 2x^2$

     The total area of the sides: $2(12x - 2x^2) + 2(18x - 2x^2) = 24x - 4x^2 + 36x - 4x^2 = 60x - 8x^2$

   Now, add the bottom area and the side areas to get the total surface area: $\text{Surface area} = (216 - 60x + 4x^2) + (60x - 8x^2)$ Simplifying: $\text{Surface area} = 216 - 4x^2$

Thus, the correct expression for the outside surface area is: $\text{Surface area} = 216 - 4x^2 \, \text{square inches}$

Let me know if you'd like more details or if you have any further questions!

Here are five follow-up questions:

1. What would happen to the volume if the cut squares were larger or smaller?
2. How does the outside surface area change as the size of the cut squares increases?
3. What is the maximum volume that this box could achieve?
4. Can you find the value of $x$ that maximizes the volume of the box?
5. How would you modify the equations if the box had a top?

Tip: Always double-check your expanded expressions to ensure that no terms were missed or miscalculated!