Answer
Solution set = $\{5\}$
Work Step by Step
For the equation to be defined, logarithms must have positive arguments
$\begin{array}{ll}
x+4 \gt 0, & \\
x \gt -4 & (*)
\end{array}$
$(*)$ is the condition eventual solutions must satisfy.
$ 2\log_{3}(x+4)=\log_{3}9+2\qquad $... LHS: power rule, RHS: $\log_{a}a^{n}=n $
$\log_{3}(x+4)^{2}=\log_{3}9+\log_{3}3^{2}\qquad $ ... RHS: product rule
$\log_{3}(x+4)^{2}=\log_{3}(9\cdot 3^{2})\qquad $ ... if $\log_{b}M=\log_{b}N,$ then M=N
... logarithmic functions are one to one,
$(x+4)^{2}=81$
$ x+4=\pm 9$
$ x=-4\pm 9$
$ x=-13\qquad $ ... does not satisfy (*) - not a solution
$ x=5\qquad $ ... satisfies (*)
Solution set = $\{5\}$
