Use the pair of functions to find $f(g(x))$ and $g(f(x))$. Simplify your answers. \[ \begin{array}{l} f(x)=x^{2}+2, \quad g(x)=\sqrt{x+8} \\ =X+10 \end{array} \] \[ f(g(x))=X+10 \] \[ g(f(x))= \]

Step 1 ：The problem is asking for the composition of two functions, \(f(g(x))\) and \(g(f(x))\).

Step 2 ：For \(f(g(x))\), we need to substitute \(g(x)\) into \(f(x)\).

Step 3 ：For \(g(f(x))\), we need to substitute \(f(x)\) into \(g(x)\).

Step 4 ：Let's start with \(f(g(x))\).

Step 5 ：Given that \(f(x) = x^2 + 2\) and \(g(x) = \sqrt{x + 8}\), we substitute \(g(x)\) into \(f(x)\) to get \(f(g(x)) = (\sqrt{x + 8})^2 + 2\).

Step 6 ：Simplifying this gives \(f(g(x)) = x + 10\).

Step 7 ：Now let's find \(g(f(x))\).

Step 8 ：We substitute \(f(x)\) into \(g(x)\) to get \(g(f(x)) = \sqrt{(x^2 + 2) + 8}\).

Step 9 ：Simplifying this gives \(g(f(x)) = \sqrt{x^2 + 10}\).

Step 10 ：So, the final answers are \(f(g(x)) = \boxed{x + 10}\) and \(g(f(x)) = \boxed{\sqrt{x^2 + 10}}\).