Answer
$f(x)=-(x+2)^2$
Work Step by Step
Given the standard function, $
f(x)=x^2
$, the equation of the graph that shifts to the left has a positive constant added to the $x-$variable. Hence, a shift of $2$ units to the left has the equation,
\begin{array}{l}\require{cancel}
f(x)=(x+2)^2
.\end{array}
Given the function, $
f(x)=(x+2)^2
$, the equation of the graph that is reflected about the $x-$axis has a negative multiplier in the $y-$variable. Hence, the equation becomes
\begin{array}{l}\require{cancel}
-f(x)=(x+2)^2
\\\\
f(x)=-(x+2)^2
.\end{array}