Example 10 Find the minimum value of the function $y=\sqrt{x^{2}+2 x+2}+$ $\sqrt{x^{2}-2 x+2}$.
Solution 1 (Construction Method): $y=\sqrt{(x+1)^{2}+1^{2}}+$ $\sqrt{(x-1)^{2}+1^{2}}$, represents the sum of the distances from a moving point $P(x, 1)$ to fixed points $A(1,0)$ and $B(-1,0)$, hence $y_{\min }=2 \sqrt{2}$.

Solution 2 (Inequality Method): $y=\sqrt{x^{2}+2 x+2}+$ $\sqrt{x^{2}-2 x+2} \geqslant 2 \sqrt[4]{\left(x^{2}+2 x+2\right)\left(x^{2}-2 x+2\right)}$ $=2 \sqrt[4]{x^{4}+4} \geqslant 2 \sqrt{2}$, when $x=0$, both equalities hold simultaneously, hence $y_{\min }=2 \sqrt{2}$.