The area of the parallelogram is$4\sqrt{29}$  sq.units If one of adjacent side is $3\overline{\mathrm{i}}+4\overline{\mathrm{j}}$ then the other side is

$$\begin{gathered} Given one adjacent side of paralle\log ram \overline{a} =3\overline{i}+4\overline{j} \\ let other side be \overline{b} \\ Given area of paralle\log ram 4\sqrt{29} \\ \Rightarrow |\overline{\mathrm{a}}\times b|=4\sqrt{29} \\ Now by verifying all options only option D satisfy the equation \left(1\right) as follows \\ let \overline{b}=2\overline{i}-4\overline{k} \end{gathered}$$ 

 $$\begin{gathered} \Rightarrow \overline{\mathrm{a}}\times \overline{\mathrm{b}}=\left|\begin{array}{ccc}\overline{\mathrm{i}}& \overline{\mathrm{j}}& \overline{\mathrm{k}}\\ 3& 4& 0\\ 2& 0& -4\end{array}\right| \\ =\overline{i}(-16)-\overline{j}(-12)+\overline{k}(-8) \\ =-16\overline{i}+12\overline{j}-8\overline{k} \\ \Rightarrow |\overline{\mathrm{a}}\times \overline{\mathrm{b}}|=\sqrt{{(-16)}^2+{\left(12\right)}^2+{(-8)}^2} \\ =\sqrt{464} \\ =4\sqrt{29} \end{gathered}$$