The range of $f\left(x\right)={\log }_3\left(5-4x-x^2\right)$ is

$\begin{array}{r}f\left(x\right)={\log }_3\left(5-4x-x^2\right)={\log }_3\left\{9-(x-2{)}^2\right\}\\ f\left(x\right)\text{ is defined if }\\ \begin{array}{rl} & 9-(x-2{)}^2>0\text{ but }9-(x-2{)}^2\le 9\\ & \text{ or }0<9-(x-2{)}^2\le 9\\ & \text{ or }-\mathrm{\infty }<{\log }_3\left\{9-(x-2{)}^2\right\}\le {\log }_{3}9\end{array}\end{array}$

$\text{ Hence, the range is }(-\mathrm{\infty },2]\text{ . }$