Example 15 Simplify $m=\frac{a^{2}(x-b)(x-c)}{(a-b)(a-c)}$ $+\frac{b^{2}(x-c)(x-a)}{(b-c)(b-a)}+\frac{c^{2}(x-a)(x-b)}{(c-a)(c-b)}$.
Solution: Let $f(x)=m-x^{2}$, then $f(x)$ is a polynomial in $x$ of degree not exceeding 2, and $f(a)=f(b)$ $=f(c)=0$, which indicates that $a$, $b$, and $c$ are all roots of $f(x)=0$. Since $f(x)$ has a degree not exceeding 2, $f(x)=0$ holds true constantly. Therefore, $m=x^{2}$.