In the given figure, we have two squares. The side of the larger one measures $a+b$ and the side of the smaller one $a$. What is the area of the highlighted gray region?
(a) $b$

(b) $a+b$

(c) $a^{2}+2 a b$

(d) $b^{2}$

(e) $2 a b+b^{2}$

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-003.jpg?height=362&width=514&top_left_y=276&top_left_x=1368)
The correct option is (e).

Solution 1: The area of a square with side length $l$ is $l^{2}$, and the area of the gray region is the difference between the areas of the larger and smaller squares. The side length of the larger square is $a+b$, so its area is $(a+b)^{2}=a^{2}+2 a b+b^{2}$. The side length of the smaller square is $a$, so its area is $a^{2}$. Therefore, the area of the gray region is $(a+b)^{2}-a^{2}=a^{2}+2 a b+b^{2}-a^{2}=2 a b+b^{2}$.

Solution 2: The area of a rectangle is the product of its width and length. According to the problem, the width of the gray region is $(a+b)-a=b$.

By dividing the gray region into two rectangles, one with width $b$ and length $a$, and the other with width $b$ and length $a+b$ (see figure), we see that the area of the gray region is the sum of the areas of these two rectangles, that is,

$$
\begin{aligned}
a \times b+b \times(a+b) & =a b+a b+b^{2} \\
& =2 a b+b^{2}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-037.jpg?height=492&width=609&top_left_y=899&top_left_x=1223)

Therefore, the area of the gray region is $2 a b+b^{2}$.

Solution 3: The gray region is formed by two rectangles with dimensions $a \times b$ and one square with side length $b$. Thus, its area is $2 a b+b^{2}$.