Answer
$g(x)=-\sqrt {x+4}-2$
Work Step by Step
Based on the final graph given, start from $f(x)=\sqrt x$ (red), we can do the following steps: shift left 4 units, reflect around the x-axis, then shift down 2 units to get $g(x)=-\sqrt {x+4}-2$ (blue) as shown in the figure.