Find maximum value of $f\left(x\right)=\left|\begin{array}{ccc}1+{\sin }^{2}x& {\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& 1+{\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+4\sin 2x\end{array}\right|$

$\because f\left(x\right)=\left|\begin{array}{ccc}1+{\sin }^{2}x& {\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& 1+{\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+4\sin 2x\end{array}\right|$
Applying $R_2\to R_2-R_1$ and $R_3\to R_3-R_1$, then
$f\left(x\right)=\left|\begin{array}{ccc}1+{\sin }^{2}x& {\cos }^{2}x& 4\sin 2x\\ -1& 1& 0\\ -1& 0& 1\end{array}\right|$
Applying $C_2\to C_2+C_1$, then

$f\left(x\right)=\left|\begin{array}{ccc}1+{\sin }^{2}x& 2& 4\sin 2x\\ -1& 0& 0\\ -1& -1& 1\end{array}\right|$
Expanding along $R_2$, then
$f\left(x\right)=\left|\begin{array}{cc}2& 4\sin 2x\\ -1& 1\end{array}\right|=2+4\sin 2x$
$\therefore$ Maximum value of
$$\begin{gathered} f\left(x\right)=2+4\left(1\right) \\ f\left(x\right)=6 \end{gathered}$$