$x^3-3 x^2+3 x+1=0$
$\begin{aligned} & \alpha+\beta+\gamma=3 ; \alpha \beta+\beta \gamma+\gamma \alpha=3 ; \alpha \beta \gamma=-1 \\ & \Rightarrow \alpha^2 \beta^2+\beta^2 \gamma^2+\gamma^2 \alpha^2 \\ & =(\alpha \beta+\beta \gamma+\gamma \alpha)^2-2\left(\alpha \gamma \beta^2+\beta \alpha^2 \gamma+\alpha \beta \gamma^2\right) \\ & =(\alpha \beta+\beta \gamma+\gamma \alpha)^2-2(\alpha \beta \gamma)(\beta+\alpha+\gamma) \\ & =(3)^2-2(-1)(3)=15\end{aligned}$