Given, $\left|\begin{array}{lll}4-x & 4+x & 4+x \\ 4+x & 4-x & 4+x \\ 4+x & 4+x & 4-x\end{array}\right|=0$
$$
\Rightarrow\left|\begin{array}{ccc}
12+x & 12+x & 12+x \\
4+x & 4-x & 4+x \\
4+x & 4+x & 4-x
\end{array}\right|=0 \quad\left[\because R_1 \rightarrow R_1+R_2+R_3\right]
$$
$$
\begin{aligned}
&\Rightarrow(12+x)\left|\begin{array}{ccc}
1 & 1 & 1 \\
4+x & 4-x & 4+x \\
4+x & 4+x & 4-x
\end{array}\right|=0 \\
&\Rightarrow(12+x)\left|\begin{array}{ccc}
0 & 0 & 1 \\
0 & 8 & 4+x \\
2 x & 8 & 4-x
\end{array}\right|=0
\end{aligned}
$$
$$
\begin{aligned}
{\left[\because \mathrm{C}_1 \rightarrow \mathrm{C}_1-\mathrm{C}_3 \text { and } \mathrm{C}_2 \rightarrow \mathrm{C}_2+\mathrm{C}_3\right] } \\
\Rightarrow(12+\mathrm{x})[1 \cdot(-16 \mathrm{x})] &=0
\end{aligned}
$$