We have,
$x^2+p x+1$ is a factor of $a x^3+b x+c=0$
$
\therefore a x^3+b x+c=\left(x^2+p x+1\right)(a x+\alpha)
$
$
\Rightarrow a x^3+b x+c=a x^3+(p a+\alpha) x^2+(p \alpha+a) x+\alpha
$
Equating the coefficient of $x^3, x^2, x$ and constant term, we get
$
\begin{aligned}
p a+\alpha & =0, & p \alpha+a=b, & \alpha=c \\
p & =\frac{-c}{a} & & {[\because \alpha=c] }
\end{aligned}
$
Putting the values of $p$ and $\alpha$ in $p \alpha+a=b$
$
\begin{aligned}
\left(-\frac{c}{a}\right)(c)+a & =b \\
-c^2+a^2 & =a b \Rightarrow a^2-c^2=a b
\end{aligned}
$