$\begin{aligned} & \text { } \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+8 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=7 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+3 \hat{\mathrm{k}} \\ & (2 \overrightarrow{\mathrm{a}}-3 \overrightarrow{\mathrm{b}}) \times(4 \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})=x \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}\end{aligned}$
$$
\begin{aligned}
& 2 \vec{a}-3 \vec{b}=-17 \hat{i}+5 \hat{j}+7 \hat{k} \\
& 4 \vec{a}+\vec{b}=15 \hat{i}-25 \hat{j}+35 \hat{k}
\end{aligned}
$$
Now, $(2 \overrightarrow{\mathrm{a}}-3 \overrightarrow{\mathrm{b}}) \times(4 \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})=\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ -17 & 5 & 7 \\ 15 & -25 & 35\end{array}\right|$
$$
\begin{aligned}
& =\hat{\mathrm{i}}(175+175)-\hat{\mathrm{j}}(-595-105)+\hat{\mathrm{k}}(25 \times 17-75) \\
& \Rightarrow(2 \overrightarrow{\mathrm{a}}-3 \overrightarrow{\mathrm{b}}) \times(4 \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})=350 \hat{\mathrm{i}}+700 \hat{\mathrm{j}}+350 \hat{\mathrm{k}}
\end{aligned}
$$
From eqns. (i) \& (ii)
$$
\begin{aligned}
& x=350, y=700 \& z=350 \\
& \therefore x+y+z=350+700+350=140
\end{aligned}
$$