\[4\sqrt{3} - 3\sqrt{27} + 2\sqrt{243} =\]
\[= 4\sqrt{3} - 3 \cdot 3\sqrt{3} + 2 \cdot 9\sqrt{3} =\]
\[= 4\sqrt{3} - 9\sqrt{3} + 18\sqrt{3} = 13\sqrt{3};\]
\[\left( 13\sqrt{3} \right)^{2} = 169 \cdot 3 = 507.\]
\[Ответ:507.\]