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the length side of the original big square is x, the length side of the 4 smaller squares is 1/5x

To solve the problem, we need to find the side length \(x\) of the original large square piece of cardboard that Alejandro used to create the boxes. Let's go through the steps:

### Problem Analysis

1. **Given Information:**
   - The large square has a side length of \(x\).
   - Each of the four smaller squares that are cut out has a side length of \(\frac{x}{5}\).
   - The resulting area of the folded box (without the flaps) is 84 cm².

2. **Objective:**
   - Determine the length \(x\) of the original square's side.

### Step-by-Step Solution

1. **Calculate the size of the cut-out squares:**
   - Each cut-out square has a side length of \(\frac{x}{5}\).

2. **Determine the dimensions of the resulting square (without flaps):**
   - After cutting out the four smaller squares, the length of the side of the remaining square part of the cardboard is:
     \[
     \text{Remaining length} = x - 2 \times \frac{x}{5} = x - \frac{2x}{5} = \frac{5x - 2x}{5} = \frac{3x}{5}.
     \]

3. **Relate the remaining area to the given area:**
   - The area of the resulting square is:
   \[
   \left(\frac{3x}{5}\right)^2 = 84 \, \text{cm}^2.
   \]

4. **Solve for \(x\):**
   \[
   \left(\frac{3x}{5}\right)^2 = 84.
   \]
   \[
   \frac{9x^2}{25} = 84.
   \]
   \[
   9x^2 = 84 \times 25.
   \]
   \[
   9x^2 = 2100.
   \]
   \[
   x^2 = \frac{2100}{9}.
   \]
   \[
   x^2 = 233.33\ldots.
   \]
   \[
   x = \sqrt{233.33\ldots}.
   \]
   \[
   x \approx 15.27.
   \]

### Answer Verification

Looking at the multiple-choice answers provided:

- **A) 10 cm**
- **B) 12 cm**
- **C) 15 cm**
- **D) 20 cm**

The closest answer to \(x \approx 15.27\) is **C) 15 cm**.

### Conclusion

The side length of the original square cardboard is approximately **15 cm**.

Would you like further details or have any questions?

### Related Questions:

1. How would the solution change if the area occupied by the box were different?
2. What would be the side length of each cut-out square if the original side length \(x\) were known?
3. How do we compute the perimeter of the final square piece after the cut-out?
4. What is the significance of understanding transformations in geometry?
5. How does changing the size of the cut-out squares affect the overall box volume?

### Tip:

When solving geometry problems involving cut-outs and transformations, always start by analyzing how the cuts impact the overall dimensions and area.

Learn how to determine the side length of the original large square and the resulting area after cutting out four smaller squares. The problem involves geometric concepts and quadratic equations. Suitable for high school students.

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