Example 3 The polynomial $\left(x^{2}+2 x+2\right)^{2001}+\left(x^{2}-\right.$ $3 x-3)^{2001}$, after expansion and combining like terms, the sum of the coefficients of the odd powers of $x$ is

保留源文本的换行和格式，直接输出翻译结果如下：

```
Example 3 The polynomial $\left(x^{2}+2 x+2\right)^{2001}+\left(x^{2}-\right.$ $3 x-3)^{2001}$, after expansion and combining like terms, the sum of the coefficients of the odd powers of $x$ is
```
Solution: Let $f(x)=\left(x^{2}+2 x+2\right)^{2001}+\left(x^{2}\right.$
$$
\begin{aligned}
& -3 x-3)^{2001} \\
= & a_{0}+a_{1} x+a_{2} x^{2}+\cdots \\
& +a_{4001} x^{4001}+a_{4002} x^{4002} .
\end{aligned}
$$

Let $x=1$ and $x=-1$, we get
$$
\begin{array}{l}
a_{0}+a_{1}+a_{2}+\cdots+a_{4001}+a_{4002} \\
=f(1)=0, \\
a_{0}-a_{1}+a_{2}-\cdots-a_{4001}+a_{4002} \\
=f(-1)=2 .
\end{array}
$$

Subtracting the two equations, we get
$$
\begin{array}{l}
2\left(a_{1}+a_{3}+\cdots+a_{4001}\right)=-2 . \\
\therefore a_{1}+a_{3}+\cdots+a_{4001}=-1 .
\end{array}
$$