$\begin{aligned} & b^2 \sin 2 C+c^2 \sin 2 B \\ & =2 b^2 \sin C \cos C+2 c^2 \sin B \cos B \\ & =\frac{2 b^2 c \cos C}{2 R}+\frac{2 c^2 b \cos B}{2 R} \\ & \qquad\left[\because \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2 R\right] \\ & =\frac{b c}{R}(b \cos C+c \cos B)=\frac{a b c}{R} \\ & {[\because \text { by projection formula } b \cos C+c \cos B=a]} \\ & =4 \Delta \\ & \quad\left[\because R=\frac{a b c}{4 \Delta}\right]\end{aligned}$