Answer
$-80 \pi$
Work Step by Step
When the surface is orientable then the flux through a surface is defined as: $\iint_S F \cdot dS=\iint_S F \cdot n dS$
where, $n$ is the unit vector.
Since, we have $\iint_S f(x,y,z) dS \approx \Sigma_{i=1}^n f[\overline{x}, \overline{y}, \overline{z}] AS_i$
Since, $\iint_S f(x,y,z) dS =g(\sqrt {x^2+y^2+z^2})$
This implies that $g(\sqrt 4) =g(2)=-5$
Now, we have $\iint_S f(x,y,z) dS =-5 \times \iint_S dS$
The integral $\iint_S dS$ shows the surface area of the sphere $=4 \pi r^2=4 \times \pi \times (4)=16 \pi$
Thus, we have $\iint_S f(x,y,z) dS =-5 \times 16 \pi=-80 \pi$
