$\begin{aligned} & \sin ^2 x+\left(3-(x-1)^2\right) \sin x-3(x-1)^2=0 \\ & \sin ^2 x+3 \sin x-(x-1)^2 \sin x-3(x-1)^2=0 \\ & \left.\sin x(\sin x+3)-(x-1)^2\right)[\sin x+3]=0 \end{aligned}$ There are two intersections between this graph. So, Number of solution will be 2 .