Example 1.17 Find the coefficient of $x^{5}$ in the expansion of $\left(1+2 x+3 x^{2}+4 x^{3}\right)^{5}$.
Solution: Since $\left(1+2 x+3 x^{2}+4 x^{3}\right)^{5}$
$$
\begin{aligned}
= & {\left[(1+x)^{2}+2 x^{2}(1+2 x)\right]^{5} \text { this is a clever factorization } } \\
= & (1+x)^{10}+10 x^{2}(1+x)^{8}(1+2 x)+ \\
& 40 x^{4}(1+x)^{6}(1+2 x)^{2}+ \\
& 80 x^{6}(1+x)^{4}(1+2 x)^{3}+\cdots,
\end{aligned}
$$

Therefore, the coefficient of $x^{5}$ in the expansion of $\left(1+2 x+3 x^{2}+4 x^{3}\right)^{5}$ is
$$
\begin{aligned}
\binom{10}{5} & +10\left[\binom{8}{3}+\binom{8}{2} \cdot 2\right]+40\left[\binom{6}{1}+2 \cdot 2\right] \\
& =252+1120+400=1772
\end{aligned}
$$