$\begin{aligned} & \text { As } A \operatorname{adj} A=|A| I, \operatorname{det}(\lambda A)=\lambda^n \operatorname{det} A \\\\ & \operatorname{det}(3 \operatorname{adj}(-6 \operatorname{adj}(3 A)))=3^3 \operatorname{det}(\operatorname{adj}(-6 \operatorname{adj}(3 A))) \\\\ & =3^3(-6 \operatorname{adj}(3 A))^2 \\\\ & =3^3(-6)^6|3 A|^4 \\\\ & =3^9 2^6 \cdot 3^{12} \cdot(-2)^4 \\\\ & =3^{21} \cdot 2^{10}\end{aligned}$

Now comparing with given condition

$$ \begin{aligned} & 2^{m+n} 3^{m n}=2^{10} \cdot 3^{21} \\\\ & m+n=10, m n=21 \\\\ & \Rightarrow m=7, n=3(m>n) \\\\ & \therefore 4 m+2 n=28+6=34 \end{aligned} $$