$4 \cdot 38 \quad$ For the equation
$$x^{3}+a x^{2}+b x+c=0$$

to have three different real roots that form a geometric progression, the real numbers $a, b, c$ should satisfy what conditions?
[Solution] Let $x_{1}, x_{1} q, x_{1} q^{2}$ be the three roots of equation (1), where $x_{1}, q$ are real numbers, $x_{1} \neq 0, q \neq 0, \pm 1$. These three roots satisfy the requirements of the problem. According to the relationship between roots and coefficients, we have
$$\left\{\begin{array}{l}
x_{1}+x_{1} q+x_{1} q^{2}=-a \\
x_{1}^{2} q+x_{1}^{2} q^{2}+x_{1}^{2} q^{3}=b \\
x_{1}^{3} q^{3}=-c
\end{array}\right.$$

For simplicity, let $m$ be the unique real number such that $m^{3}=-c$. Then (2) becomes
$$\left\{\begin{array}{l}
x_{1}\left(1+q+q^{2}\right)=-a \\
x_{1}^{2} q\left(1+q+q^{2}\right)=b \\
x_{1} q=m
\end{array}\right.$$

This system of equations is equivalent to the system
$$\left\{\begin{array}{l}
x_{1}\left(1+q+q^{2}\right)=-a \\
a x_{1} q+b=0 \\
x_{1} q=m
\end{array}\right.$$

From the last two equations of (4), we get
$$b=-a m$$

Thus, we have obtained the first necessary condition for the coefficients of the original equation (1). If this condition is satisfied, (4) is equivalent to
$$\left\{\begin{array}{l}
x_{1}\left(1+q+q^{2}\right)=-a \\
x_{1} q=m .
\end{array}\right.$$

In the first equation of (5), substituting $m$ for $x_{1} q$, we get
$$\left\{\begin{aligned}
x_{1}+m q & =-a-m, \\
x_{1} q & =m .
\end{aligned}\right.$$
(5) is equivalent to (6).

Noting the characteristics of $x_{1}, q$, the second necessary condition for the coefficients $a, b, c$ is
$$m \neq 0$$

From (6), eliminating $x_{1}$, we get the equation
$$m q^{2}+(m+a) q+m=0$$

Since $q$ is a real number, it follows that
$$(m+a)^{2}-4 m^{2} \geqslant 0$$

That is,
$$a^{2}+2 a m-3 m^{2} \geqslant 0$$

Since $q \neq \pm 1$, from equation (7), we get the other two necessary conditions for the coefficients $a, b, c$:
$$\begin{array}{l}
a \neq-3 m \\
a \neq m
\end{array}$$

Note that when $a=-3 m$ or $a=m$, the value of the trinomial $a^{2}+2 a m-3 m^{2}$ is 0. This allows us to replace conditions ( $(\mathcal{)},(\delta)$ and (E) with a single condition:
$$a^{2}+2 a m-3 m^{2}>0$$

Thus, we have obtained the three necessary conditions (a), (B), (7) that the coefficients $a, b, c$ of equation (1) should satisfy.

The conditions are also sufficient. That is, if conditions (B), (7), and (E) are satisfied, there must exist $x_{1}, q$ that meet the requirements.

In fact, from conditions (B) and (7), we can get that equation (7) has two real roots. Let $q$ be one of them (then the other root is $1 / q$). By condition (3), $q \neq 0$; by condition (7), $q \neq \pm 1$. Thus, from the second equation of (6), we get $x_{1}=m / q$, and $x_{1}$ is a real number that satisfies the first equation of (6). Moreover, by condition (B), $x_{1} \neq 0$. If condition (E) is satisfied, then (6) is equivalent to (2). Thus, the values of $x_{1}$ and $q$ obtained satisfy (1), which shows that $x_{1}, x_{1} q, x_{1} q^{2}$ are roots of (1).

Condition (7) means that $m$ lies between $a$ and $-a / 3$. Since $m^{3}=-c$, the conditions (B), (7), and (E) can be rewritten as follows:
$$\left\{\begin{array}{l}
b^{3}=a^{3} c, \\
c \neq 0, \\
c \text { lies between }-a^{3} \text { and } a^{3} / 27.
\end{array}\right.$$

This is the required condition.