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To find the area of the shaded region \( S \), we need to calculate the area of the square and the areas of the quarter circles, then find their intersections.

The square has a side length of 4 units, so its area is:
\[ A_{\text{square}} = 4 \times 4 = 16 \text{ square units} \]

Each quarter circle has a radius of 4 units, so the area of one quarter circle is:
\[ A_{\text{quarter circle}} = \frac{1}{4} \pi (4)^2 = 4\pi \text{ square units} \]

There are two quarter circles, so their combined area is:
\[ A_{\text{two quarter circles}} = 2 \times 4\pi = 8\pi \text{ square units} \]

The shaded region \( S \) is the intersection of these two quarter circles within the square, which is equivalent to the area of the square minus the areas of the two quarter circles, and then plus the overlapping area of the quarter circles.

The overlapping area of the two quarter circles is one-quarter of the area of a circle with radius 4 (because this overlap forms a quarter of the total area when the two circles overlap in this manner):
\[ A_{\text{overlap}} = \frac{1}{4} \pi (4)^2 = 4\pi \text{ square units} \]

Thus, the area of the shaded region \( S \) can be calculated as follows:
\[ A_S = A_{\text{square}} - (2 \times A_{\text{quarter circle}}) + A_{\text{overlap}} \]
\[ A_S = 16 - 8\pi + 4\pi \]
\[ A_S = 16 - 4\pi \]

So, the area of the shaded region \( S \) is:
\[ \boxed{16 - 4\pi \text{ square units}} \]

Do you have any questions or need further details?

Here are five questions you might want to ask next:
1. How do you calculate the area of a quarter circle?
2. Can you explain the steps to find the area of a composite figure?
3. What is the formula for the area of a circle?
4. How can we find the area of other overlapping shapes?
5. Can you provide more examples of finding areas of complex regions?

**Tip:** When dealing with composite figures, break them down into simpler shapes whose areas you can calculate easily, and then combine those areas appropriately.

Learn how to calculate the area of a shaded region within a square using quarter circles. This problem involves geometry and circle geometry, covering the formulas for the area of a square and quarter circle. Suitable for high school students.

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