Answer
$3.5\times 10^{-4}\;\rm N\cdot m^2/C$
Work Step by Step
We know that the electric flux is given by
$$ \Phi=\vec E\cdot \vec A $$
Hence,
$$\Phi=(350x+150)\;\hat i\cdot \vec A$$
The box has 6 faces but there are only two faces that the electric field enters and comes out from them since the given electric field formula is on the $x$-direction only. The other 4 faces the angle between the electric field and the area vector is 90$^\circ$ which means that $\cos\theta=0$ and hence the flux is zero as well on all of them.
The flux enters on the face where $x=0$ cm and comes out from the face where $x=1$ cm since the dimensions of the box are 1 cm $\times$ 1 cm $\times$ 1 cm.
$$\Phi=(350x+150)\;\hat i\cdot \vec A_1+(350x+150)\;\hat i\cdot \vec A_2$$
$$\Phi=(350[0]+150)\;\hat i(0.01^2)\cos180^\circ\;\hat i+(350[0.01]+150)(0.01^2)\cos0^\circ\;\hat i$$
where at $x=0$ cm, $\theta=180^\circ$ and at $x=1$ cm, $\theta=0^\circ$
$$\Phi=\color{red}{\bf 3.5\times 10^{-4}}\;\rm N\cdot m^2/C$$