Example 8 When $m$ satisfies what conditions, the equation
$$
(m+3) x^{2}-4 m x+2 m-1=0
$$

has roots of opposite signs and the absolute value of the negative root is greater than the positive root?
Explanation: For equation (1), the conditions for the roots to have opposite signs and the absolute value of the negative root to be greater than the positive root are
$$
\left\{\begin{array}{l}
m+3 \neq 0, \\
\Delta=16 m^{2}-4(m+3)(2 m-1)>0, \\
x_{1}+x_{2}=\frac{4 m}{m+3}<0, \\
x_{1} x_{2}=\frac{2 m-1}{m+3}<0 .
\end{array}\right.
$$

Solving the system of inequalities, we get $-3<m<0$.