\(2{\log _9}\left( {4{x^2} + 1} \right) \ge {\log _3}\left( {3{x^2} + 4x + 1} \right)\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\log _3}\left( {4{x^2} + 1} \right) \ge {\log _3}\left( {3{x^2} + 4x + 1} \right)\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{4{x^2} + 1 \ge 3{x^2} + 4x + 1,}\\{3{x^2} + 4x + 1 > 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{{x^2}-4x \ge 0,\,\,\,\,\,\,\,\,\,}\\{3{x^2} + 4x + 1 > 0}\end{array}\,\,\,\,\,\,\,\, \Leftrightarrow } \right.} \right.\)
\( \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x\, \in \,\left( {-\infty ;0} \right] \cup \left[ {4;\infty } \right),\,\,\,\,\,\,\,\,}\\{x\, \in \,\left( {-\infty ;-1} \right) \cup \left( {-\dfrac{1}{3};\infty } \right)}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x\, \in \,\left( {-\infty ;-1} \right) \cup \left( {-\dfrac{1}{3};0} \right] \cup \left[ {4;\infty } \right).\)
Ответ: \(\left( {-\infty ;-1} \right) \cup \left( {-\dfrac{1}{3};0} \right] \cup \left[ {4;\infty } \right).\)