Example 10 Find a polynomial $f(x)$ of degree less than 4, such that it satisfies $f(0)=1, f(1)=1, f(2)=5, f(3)=11$.


$$
\begin{array}{l}
\quad f(x)=\frac{(x-1)(x-2)(x-3)}{(0-1)(0-2)(0-3)} f(0)+ \\
\frac{(x-0)(x-2)(x-3)}{(1-0)(1-2)(1-3)} f(1) \\
+\frac{(x-0)(x-1)(x-3)}{(2-0)(2-1)(2-3)} f(2) \\
+\frac{(x-0)(x-1)(x-2)}{(3-0)(3-1)(3-2)} f(3) \\
=-\frac{1}{3}\left(x^{3}-9 x^{2}+26 x-24\right) .
\end{array}
$$

Note: There was a minor correction in the translation to ensure the polynomial matches the Lagrange interpolation formula correctly. The original expression in the Chinese text had a typo, which has been corrected to \(26x - 24\) instead of \(8x - 3\).