Let $\mathbf{a}=3 \mathbf{i}-2 \mathbf{j}-\mathbf{k}, \quad \mathbf{b}=2 \mathbf{i}+3 \mathbf{j}-4 \mathbf{k}, \quad \mathbf{c}=-\mathbf{i}+\mathbf{j}+2 \mathbf{k}$ and $\mathbf{d}=4 \mathbf{i}+5 \mathbf{j}+\lambda \mathbf{k}$.
Since the points are coplanar,
So,
$[\mathbf{d} \mathbf{b} \mathbf{c}]+[\mathbf{d} \mathbf{c} \mathbf{a}]+[\mathbf{d} \mathbf{a} \mathbf{b}]=[\mathbf{a} \mathbf{b} \mathbf{c}]$
$\begin{aligned}
& \Rightarrow\left|\begin{array}{ccc}
4 & 5 & \lambda \\
2 & 3 & -4 \\
-1 & 1 & 2
\end{array}\right|+\left|\begin{array}{ccc}
4 & 5 & \lambda \\
-1 & 1 & 2 \\
3 & -2 & -1
\end{array}\right|+\left|\begin{array}{ccc}
4 & 5 & \lambda \\
3 & -2 & -1 \\
2 & 3 & -4
\end{array}\right| \\
& =\left|\begin{array}{ccc}
3 & -2 & -1 \\
2 & 3 & -4 \\
-1 & 1 & 2
\end{array}\right|
\end{aligned}$
$\Rightarrow 40+5 \lambda+37-\lambda+94+13 \lambda=25 \Rightarrow \lambda=\frac{-146}{17} \text {. }$