Step 1: Field depends only on $x$, so flux through the left and right faces (perpendicular to x-axis) are the only non-zero contributions; flux through other 4 faces is zero (E is parallel to those faces).

Step 2: At the left face, $x=0$: $E_{left}=500(0)=0$, so flux $=0$.

Step 3: At the right face, $x=0.1$ m: $E_{right}=500(0.1)=50\ NC^{-1}$.

Step 4: Area of each face $= (0.1)^2 = 0.01\ m^2$. Flux through right face $= 50\times0.01 = 0.5\ Nm^2C^{-1}$.

Step 5: Total flux $\phi = 0.5\ Nm^2C^{-1}$ (only outward through the right face).

Step 6: By Gauss's law, $q_{enc} = \varepsilon_0\phi = 8.85\times10^{-12}\times0.5 = 4.43\times10^{-12}\ C$.