Given that, $\mathbf{a}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}}$
$$
\mathbf{b}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}
$$
$$
\mathbf{c}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}-4 \hat{\mathbf{k}} \text { and } \mathbf{d}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}
$$
Now, $\mathbf{a} \times \mathbf{b}=\left|\begin{array}{ccc}\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 2 & 1 & -3 \\ 1 & -2 & 1\end{array}\right|$
$$
\begin{aligned}
& =\mathbf{i}(1-6)-\hat{\mathbf{j}}(2+3)+\hat{\mathbf{k}}(-4-1) \\
\mathbf{a} \times \mathbf{b} & =-5 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}
\end{aligned}
$$
Now,
$$
\begin{aligned}
\mathbf{c} \times \mathbf{d} & =\left|\begin{array}{ccc}
\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\
-1 & 1 & -4 \\
1 & 1 & 1
\end{array}\right| \\
& =\hat{\mathbf{i}}(1+4)-\hat{\mathbf{j}}(-1+4)+\hat{\mathbf{k}}(-1-1) \\
& =5 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}
\end{aligned}
$$
Now,
$$
\begin{aligned}
(\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})=\left|\begin{array}{ccc}
\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\
-5 & -5 & -5 \\
5 & -3 & -2
\end{array}\right| \\
=\hat{\mathbf{i}}(10-15)-\hat{\mathbf{j}}(10+25)+\hat{\mathbf{k}}(+15+25) \\
=-5 \hat{\mathbf{i}}-35 \hat{\mathbf{j}}+40 \hat{\mathbf{k}}=5(-\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+8 \hat{\mathbf{k}})
\end{aligned}
$$
$$
\begin{aligned}
|(\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})| & =5 \sqrt{(-1)^2+(-7)^2+(8)^2} \\
& =5 \sqrt{1+49+64}=5 \sqrt{114}
\end{aligned}
$$