\[\begin{array}{l} 4 \cdot {7^{2x + 4}} - {3^{2x + 6}} - 2 \cdot {7^{2x + 3}} + {3^{2x + 3}} = 0 \Leftrightarrow 4 \cdot {7^{2x + 4}} - 2 \cdot {7^{2x + 3}} = {3^{2x + 6}} - {3^{2x + 3}} \Leftrightarrow \hfill \\ {7^{2x + 3}}\left( {4 \cdot 7 - 2} \right) = {3^{2x + 3}}\left( {{3^3} - 1} \right) \Leftrightarrow {7^{2x + 3}} = {3^{2x + 3}} \Leftrightarrow {\left( {\frac{3}{7}} \right)^{2x + 3}} = 1 \Leftrightarrow \hfill \\ 2x + 3 = 0 \Leftrightarrow x = - \frac{3}{2}. \hfill \\ \end{array}\]

\[x = - \frac{3}{2}\]