For example, the minimum value of $9 \sqrt{x^{2}-4 x+8}+\sqrt{x^{2}+2 x+2}$ is $\qquad$
Solution: The original expression $=\sqrt{(x-2)^{2}+2^{2}}+$ $\sqrt{(x+1)^{2}+(-1)^{2}}$, which represents the sum of the distances from point $(x, 0)$ to points $A(2,2)$ and $B(-1,1)$. From the previous problem, we know that the required minimum value is
$$
\sqrt{(2+1)^{2}+(2+1)^{2}}=3 \sqrt{2} \text {. }
$$