Plane passing through the line of intersection of planes $\mathbf{r} \cdot \mathbf{n}=1$ and $\mathbf{r} \cdot \mathbf{m}=-4$ $\mathbf{r} \cdot(\mathbf{n}+\lambda \mathbf{m})=1-4 \lambda($ where $\lambda$ is a scalar quantities)

This plane is $\perp$ to the plane $\mathbf{r} \cdot \mathbf{1}=-8$
So,
$$
\begin{gathered}
(\mathbf{n}+\lambda \mathbf{m}) \cdot \mathbf{l}=0 \\
\mathbf{n} \cdot \mathbf{1}+\lambda \mathbf{m} \cdot \mathbf{1}=0
\end{gathered}
$$
$$
\begin{aligned}
\Rightarrow(2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}) \cdot(2 \hat{\mathbf{i}}-\hat{\mathbf{j}} & +\hat{\mathbf{k}}) \\
& +\lambda(\hat{\mathbf{i}}-\hat{\mathbf{j}}) \cdot(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=0 \\
\Rightarrow \quad(4+3+4)+\lambda(2+1) & =0 \\
\Rightarrow \quad & \lambda=\frac{-11}{3} .
\end{aligned}
$$

So, required plane is,
$$
\begin{gathered}
\mathbf{r} \cdot\left[2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}} \frac{-11}{3}(\hat{\mathbf{i}}-\hat{\mathbf{j}})\right]=1-4\left(\frac{-11}{3}\right) \\
\Rightarrow \quad \mathbf{r} \cdot\left(\frac{-5 \hat{\mathbf{i}}}{3}+\frac{2 \hat{\mathbf{j}}}{3}+4 \hat{\mathbf{k}}\right)=\frac{47}{3} \\
(x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}}) \cdot(-5 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+12 \hat{\mathbf{k}})=47 \\
\quad[\mathbf{r}=x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}}] \\
-5 x+2 y+12 z=47 \\
5 x-2 y-12 z+47=0 .
\end{gathered}
$$