Example 1 Solve the inequalities:
(1) $(x-2)(x+1)(x+3)>0$;
(2) $(x-2)(x+1)(x+3)(x+5)>0$.
Solution (1) Solution one: The roots of the factors on the left side of the inequality are $-3$, $-1$, and $2$. None of these roots are solutions to the inequality. These roots divide the number line into four intervals. The signs of the factors and their product in these intervals can be listed as follows:
\begin{tabular}{|c|c|c|c|c|}
\hline Interval where $x$ is located & $(-\infty,-3)$ & $(-3,-1)$ & $(-1,2)$ & $(2,+\infty)$ \\
\hline $x+3$ & - & + & + & + \\
\hline $x+1$ & - & - & + & + \\
\hline $x-2$ & - & - & - & + \\
\hline $(x-2)(x+1)(x+3)$ & - & + & - & + \\
\hline
\end{tabular}

Therefore, the solution set of the original inequality is $(-3,-1) \cup(2,+\infty)$.
Solution two (Root Marking Method on Number Line): As shown in Figure 2-5, mark the roots of the factors on the left side of the inequality, $-3$, $-1$, and $2$, on the $x$-axis (number line) in ascending order from left to right, with the positive direction to the right. Then, starting from the right end, draw a curve from above the $x$-axis through the largest root 2 to below the $x$-axis, then up through the root -1 to above the $x$-axis, and down through the root -3 to below the $x$-axis. The intervals where the curve is above the $x$-axis correspond to the intervals where the product is positive, which are the solutions to the original inequality.
Therefore, the solution set of the original inequality is $(-3,-1) \cup(2,+\infty)$.
(2) (Root Marking Method on Number Line)
As shown in Figure 2-6, mark the roots of the factors on the left side of the inequality, $-5$, $-3$, $-1$, and $2$, on the $x$-axis. Then, starting from the right end, draw a curve from above the $x$-axis through the largest root 2 to below the $x$-axis, then up through the root -1 to above the $x$-axis, down through the root -3 to below the $x$-axis, and finally up through the root -5 to above the $x$-axis. The intervals where the curve is above the $x$-axis correspond to the intervals where the product is positive, which are the solutions to the original inequality.
Therefore, the solution set of the original inequality is $(-\infty,-5) \cup(-3,-1) \cup(2,+\infty)$.
Solution Guide: Only the relevant numbers are marked on the number line.