Not much to do here other than to compute each of these.

\[\begin{align*}\left( {f \circ g} \right)\left( x \right) & = f\left[ {g\left( x \right)} \right] = f\left[ {\sqrt {5 + {x^2}} } \right] = {\left( {\sqrt {5 + {x^2}} } \right)^2} + 3 = 8 + {x^2}\\ & \\ \left( {g \circ f} \right)\left( x \right) & = g\left[ {f\left( x \right)} \right] = g\left[ {{x^2} + 3} \right] = \sqrt {5 + {{\left( {{x^2} + 3} \right)}^2}} = \sqrt {{x^4} + 6{x^2} + 14} \end{align*}\]