Let $\alpha, \beta, \gamma$ are roots of equation
$$
\begin{aligned}
& x^3+a x^2+b x+c=0 \\
& \alpha+\beta+\gamma=-a \\
& \therefore \quad \alpha \beta+\beta \gamma+\gamma \alpha=b \\
& \text { Now, } \alpha^{-1}+\beta^{-1}+\gamma^{-1}=\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma} \\
&=\frac{\alpha \beta+\beta \gamma+\gamma \alpha}{\alpha \beta \gamma}=\frac{b}{-c}=-\frac{b}{c}
\end{aligned}
$$