Example 5 Let real numbers $a, b, c, d, e \geqslant -1$, and 
$$
\begin{aligned}
a+b+ 
c+d+e & =5 . \text{ Find } \\
S & =(a+b)(b+c)(c+d)(d+e)(e+a)
\end{aligned}
$$

the minimum and maximum values of $S$. ${ }^{[2]}$
(34th China Mathematical Olympiad)
【Analysis】If $e$ is the largest, then $d+e$ and $e+a$ are both greater than 0, meaning there must be two adjacent terms that are positive. Consider the other three terms.
First, find the minimum value of $S$.
At this point, it must be less than 0, and there are two cases.
(1) Two positive and three negative.
Assume $-2 \leqslant a+b, b+c, c+d < 0$.
By the AM-GM inequality,
$-S \leqslant\left(\frac{-a-2 b-2 c-d}{3}\right)^{3}\left(\frac{a+d+2 e}{2}\right)^{2}$.
Let $a+d+2 e=A$. Then
$-S \leqslant\left(\frac{A-10}{3}\right)^{3}\left(\frac{A}{2}\right)^{2}$,

where $10 \leqslant A=10-a-2 b-2 c-d \leqslant 16$.
Thus, $-S \leqslant 512, S \geqslant-512$.
(2) Four positive and one negative.
Assume $-2 \leqslant a+b < 0$.
By the AM-GM inequality,
$-S \leqslant(-a-b)\left(\frac{a+b+2 c+2 d+2 e}{4}\right)^{4} \leqslant 162$.
When $a=b=c=d=-1, e=9$, $S$ achieves its minimum value of -512.
Next, find the maximum value of $S$.
At this point, it must be greater than 0. From the above analysis, there are two cases.
(1) Five positive.
By the AM-GM inequality,
$S \leqslant\left(\frac{2 a+2 b+2 c+2 d+2 e}{5}\right)^{5}=32$.
(2) Three positive and two negative.
By symmetry, there are two sub-cases to consider.
(i) Two negatives adjacent.
Assume $-2 \leqslant a+b, b+c < 0$.
Combining the equality conditions, by the AM-GM inequality,
$S \leqslant\left(\frac{-a-2 b-c}{2}\right)^{2}\left(\frac{a+c+d+e}{2}\right)^{2}(d+e)$.
Since $-4 \leqslant -a-2 b-c < 0$,
$a+c+d+e=5-b \leqslant 6$,
$d+e=5-a-b-c \leqslant 8$,
thus $S \leqslant (-2)^{2} \times 3^{2} \times 8 = 288$.
(ii) Two negatives not adjacent (separated by one or two terms).
Assume $-2 \leqslant a+b, c+d < 0$. Then $e < 5-a-d \leqslant 7$.
By the AM-GM inequality,
$$
S \leqslant\left(\frac{2 e}{5}\right)^{5} < 288.
$$

When $a=b=c=-1, d=e=4$, $S$ achieves its maximum value of 288.