A square has sides of length $ 10$, and a circle centered at one of its vertices has radius $ 10$. What is the area of the union of the regions enclosed by the square and the circle?

$ \textbf{(A)}\ 200 \plus{} 25\pi\qquad \textbf{(B)}\ 100 \plus{} 75\pi\qquad \textbf{(C)}\ 75 \plus{} 100\pi\qquad \textbf{(D)}\ 100 \plus{} 100\pi$
$ \textbf{(E)}\ 100 \plus{} 125\pi$
1. **Calculate the area of the square:**
   The side length of the square is given as \(10\). The area \(A_{\text{square}}\) of a square is calculated using the formula:
   \[
   A_{\text{square}} = \text{side}^2 = 10^2 = 100
   \]

2. **Calculate the area of the circle:**
   The radius of the circle is given as \(10\). The area \(A_{\text{circle}}\) of a circle is calculated using the formula:
   \[
   A_{\text{circle}} = \pi \times \text{radius}^2 = \pi \times 10^2 = 100\pi
   \]

3. **Determine the overlapping area:**
   Since the circle is centered at one of the vertices of the square and has a radius equal to the side length of the square, one-quarter of the circle overlaps with the square. The area of this overlapping region is:
   \[
   A_{\text{overlap}} = \frac{1}{4} \times A_{\text{circle}} = \frac{1}{4} \times 100\pi = 25\pi
   \]

4. **Calculate the area of the union of the square and the circle:**
   To find the area of the union of the square and the circle, we add the areas of the square and the circle and then subtract the overlapping area:
   \[
   A_{\text{union}} = A_{\text{square}} + A_{\text{circle}} - A_{\text{overlap}} = 100 + 100\pi - 25\pi = 100 + 75\pi
   \]

The final answer is \(\boxed{100 + 75\pi}\)