Example 8 Solve the equation
$$
x+\sqrt{x}+\sqrt{x+2}+\sqrt{x^{2}+2 x}=3 .
$$
【Analysis and Solution】Let the function be
$$
f(x)=x+\sqrt{x}+\sqrt{x+2}+\sqrt{x^{2}+2 x} \text {. }
$$

Then $f(x)$ is an increasing function on the interval $[0,+\infty)$.
Also, by observation, $f\left(\frac{1}{4}\right)=3$, thus, the original equation has a unique solution
$$
x=\frac{1}{4} \text {. }
$$