Use the pair of functions to find $f(g(x))$ and $g(f(x))$. Simplify your answers.
$$\begin{array}{l}f(x)=x^2+8,g(x)=\sqrt{x+4}\\ f(g(x))=◻\\ g(f(x))=◻\end{array}$$

Step 1 ：Given the functions $f(x)=x^2+8$ and $g(x)=\sqrt{x+4}$, we are asked to find $f(g(x))$ and $g(f(x))$.

Step 2 ：To find $f(g(x))$, we substitute $g(x)$ into $f(x)$, so wherever we see $x$ in $f(x)$, we replace it with $g(x)$.

Step 3 ：Substituting $g(x)$ into $f(x)$, we get $f(g(x))=(\sqrt{x+4}{)}^2+8$.

Step 4 ：Simplifying this, we get $f(g(x))=x+12$.

Step 5 ：To find $g(f(x))$, we substitute $f(x)$ into $g(x)$, so wherever we see $x$ in $g(x)$, we replace it with $f(x)$.

Step 6 ：Substituting $f(x)$ into $g(x)$, we get $g(f(x))=\sqrt{(x^2+8)+4}$.

Step 7 ：Simplifying this, we get $g(f(x))=\sqrt{x^2+12}$.

Step 8 ：So, the simplified forms of $f(g(x))$ and $g(f(x))$ are $f(g(x))=\boxed{{\displaystyle x+12}}$ and $g(f(x))=\boxed{{\displaystyle \sqrt{x^2+12}}}$.