\(3.3^{n-1}(6.3^{n+2}+3)-2.3^n(3^{n+3}-1)=405\)
\(\Leftrightarrow 3^n(6.9.3^n+3)-2.3^n(27.3^n-1)=405\)
\(\Leftrightarrow 54.3^{2n}+3.3^n-54.3^{2n}+2.3^n=405\)
\(\Leftrightarrow 5.3^n=405\)
\(\Leftrightarrow3^n=81\)
\(\Leftrightarrow 3^n=3^4\)
\(\Leftrightarrow n=4\).