Answer
$$D_h= \mathbb{R} - \{ -6 \}$$
Work Step by Step
Please note that the function $\log_bM$ is defined if and only if $M \gt 0$. So for the function $h(x)=\log_3(x+6)^2$ we have$$(x+6)^2 \gt 0 \quad \Rightarrow \quad x \in \mathbb{R}- \{ -6 \}.$$Thus, the domain of the function is$$D_h= \mathbb{R}- \{ -6 \} .$$
