We will use that $(a+b)^{2}=a^{2}+2 a b+b^{2}$. Remember that the area of a square with side $l$ is $l^{2}$. Note that the area of the gray region is the difference between the areas of the larger and the smaller square. The side of the larger one is $a+b$, so its area is $(a+b)^{2}$. The side of the smaller one is $a$, so its area is $a^{2}$. We conclude that the area of the gray region is: $(a+b)^{2}-a^{2}=a^{2}+2 a b+b^{2}-a^{2}=2 a b+b^{2}$.
2 - Remember that the area of a rectangle is the product of its width and length.

We can divide the gray region into two rectangles, one with width $b$ and length $a$, and the other with width $b$ and length $a+b$, as shown in the figure. The gray area is the sum of the areas of these two rectangles, that is:

$a \times b + b \times (a + b) = ab + ab + b^2 = 2ab + b^2$.

Therefore, the requested area is $2ab + b^2$.

![](https://cdn.mathpix.com/cropped/2024_05_01_8f3e432678bf01e6bca5g-05.jpg?height=434&width=548&top_left_y=388&top_left_x=1325)

Solution 3 - The gray region is formed by 2 rectangles with dimensions $a \times b$ and one square with side $b$. Therefore, its area is: $2 \times ab + b^2$.