$\vec{a}=2 \hat{i}-\hat{j}+3 \hat{k}, \vec{b}=-3 \hat{i}+5 \hat{j}-4 \hat{k}, \vec{c}=6 \vec{i}-4 \hat{j}+5 \hat{k}$
$\overrightarrow{\mathrm{a}} \times \hat{\mathrm{b}}\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ 2 & -1 & 3 \\ -3 & 5 & -9\end{array}\right|=\hat{\mathrm{i}}(4-15)-\hat{\mathrm{j}}(-8+9)+\hat{\mathrm{k}}(10-3)$
$=-11 \hat{\mathrm{i}}-\hat{\mathrm{j}}+7 \hat{\mathrm{k}}$
$\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ -3 & 5 & -4 \\ 6 & -4 & 5\end{array}\right|=\hat{\mathrm{i}}(25-16)-\hat{\mathrm{j}}(-15+24)+\hat{\mathrm{k}}(12-30)$
$=9 \hat{i}-9 \hat{j}-18 \hat{k}$
$\begin{aligned} & (\vec{a} \times \vec{b}) \cdot(\vec{b} \times \vec{c})=(-11 \hat{i}-\hat{j}+7 \hat{k}) \cdot(9 \hat{i}-9 \hat{j}-18 \hat{k}) \\ & =-99+9-126=-216\end{aligned}$