15. (15 points) Two identical squares overlap. The upper square is moved 3 cm to the right and then 5 cm downward, resulting in the figure shown. It is known that the area of the shaded part is 57 square centimeters. Find the side length of the square.


【Analysis】As shown in the figure, divide the shaded area into two rectangles. One rectangle has a length equal to the side length of the square and a width of 3 cm. The other rectangle has a length equal to the side length of the square minus 3 cm and a width of 5 cm. Let the side length of the square be $a$. According to the area formula for rectangles, express the areas of the two rectangles respectively, and then add them together to get the area of the shaded part, which is 57. From this, set up an equation to find the side length of the square.
【Solution】Solution: Let the side length of the square be $a$ cm, then:
$$
\begin{aligned}
3 a+(a-3) \times 5 & =57 \\
3 a+5 a-15 & =57 \\
8 a & =72 \\
a & =9
\end{aligned}
$$

Answer: The side length of the square is 9 cm.