Example 2 Let $f: \mathbf{R} \rightarrow \mathbf{R}$ satisfy
$$
(x+y)(f(x)-f(y))=(x-y) f(x+y) \text {. }
$$

Find the analytical expression of the function $f(x)$. ${ }^{[1]}$
(2012, Singapore Mathematical Olympiad)
【Analysis】Notice that, the equation has a "cyclic" effect, $x+y$ appears as a coefficient on the left side, and on the right side it becomes $f(x+y)$, the other part on the left is $f(x)-f(y)$, while on the right side it becomes $x-y$, and so on. This cyclic nature allows us to construct a system of single equations, thereby solving the problem.
Let $y=1$. Then
$$
(x+1)(f(x)-f(1))=(x-1) f(x+1) \text {. }
$$

To achieve the goal, we first construct a system of equations involving $f(x)$ and $f(x+1)$, and then eliminate terms to make $f(x+1)$ appear on the right side, and consider how to make $f(x+1)$ appear on the left side while $f(x)$ appears on the right side.
Therefore, let $x=x+1, y=-1$. Then
$$
x(f(x+1)-f(-1))=(x+2) f(x) \text {. }
$$

Simplifying equations (1) and (2) yields
$$
f(x)=\frac{f(1)+f(-1)}{2} x^{2}+\frac{f(1)-f(-1)}{2} x \text {. }
$$

Therefore, for all real numbers $a$ and $b$, we have
$$
f(x)=a x^{2}+b x \text {. }
$$

Observing the form of the problem, tracing back to its origin, and decomposing the problem in the same way it was formed, is an important method to open up one's thinking.