[tex]\it 3^{2n+2}\cdot4^{2n+3}-2^{2n+1}\cdot6^{2n+3} =3\cdot3^{2n+1}\cdot4^2\cdot4^{2n+1} -
\\\;\\
-\it2^{\it2n+1}\cdot \it6^2\cdot \it6^{\it2n+1} = 48\cdot12^{2n+1}-36\cdot12^{2n+1} =12^{2n+1}(48-36) =
\\\;\\
=12^{\it2n+1}\cdot \it12=\it12^{2n+2} = (12^{n+1})^2
[/tex]
