Example 3 Given
$$
a^{2}(b+c)=b^{2}(a+c)=2010 \text {, and } a \neq b \text {. }
$$

Then $c^{2}(a+b)=$ $\qquad$ [2]
$(2010$, I Love Mathematics Junior High School Summer Camp Mathematics Competition)
【Analysis】The given condition equation has the same structure as the algebraic expression to be evaluated. According to the known condition equation, it is impossible to determine the values of $a$, $b$, and $c$. We can only conjecture that there is an intrinsic relationship between $a$, $b$, and $c$. By constructing a new polynomial through subtraction and factoring, we can find their relationship, which is that they are equal, thus solving the evaluation of the unknown polynomial.
Solution Note that,
$$
\begin{array}{l}
a^{2}(b+c)-b^{2}(a+c) \\
=(a-b)(a b+b c+c a)=0 .
\end{array}
$$

Since $a \neq b$, then $a b+b c+c a=0$.
Therefore, $c^{2}(a+b)-b^{2}(a+c)$
$$
=(c-b)(a b+b c+c a)=0 \text {. }
$$

Thus, $c^{2}(a+b)=b^{2}(a+c)=2010$.