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

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 \(f(g(x)) = \boxed{x + 12}\) and \(g(f(x)) = \boxed{\sqrt{x^{2} + 12}}\).