Area of quadrilateral $$A B C D=$$ area of $$\triangle A B C+$$ area of $$\triangle A D C$$

In $$\triangle A B C$$

$$\begin{aligned} & \overrightarrow{A B}=4,8,-8 \\ & \overrightarrow{B C}=-2,-3,-4 \\ & \overrightarrow{A B} \times \overrightarrow{B C}=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 4 & 8 & -8 \\ -2 & -3 & -4 \end{array}\right| \\ & =-56 \hat{i}+32 \hat{j}+4 \hat{k} \end{aligned}$$

Area of $$\triangle A B C=\frac{1}{2}|\overrightarrow{A B} \times \overrightarrow{B C}|$$

$$\begin{aligned} & =\frac{1}{2} \sqrt{56^2+32^2+4^2} \\ & =\frac{1}{2} \sqrt{4176}=\frac{12 \sqrt{29}}{2}=6 \sqrt{29} \end{aligned}$$

In $$\triangle A D C=\overrightarrow{A D}=-2,-3,-4$$

$$\overrightarrow{D C}=4,8,-8$$

$$\overrightarrow{A D} \times \overrightarrow{D C}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ -2 & -3 & -4 \\ 4 & 8 & -8\end{array}\right|$$

$$=56 \hat{i}-32 \hat{j}-4 k$$

$$\text { Area of } \frac{1}{2}|\overrightarrow{A D} \times \overrightarrow{D C}|$$

$$\begin{aligned} & =\frac{1}{2} \sqrt{4176} \\ & =6 \sqrt{29} \end{aligned}$$

Area of $$A B C D=6 \sqrt{29}+6 \sqrt{29}$$

$$=12 \sqrt{29}$$