I guys, hope someone can help me out with this problem.
You are correct up to $\displaystyle \displaystyle \begin{align*} m = \frac{3 - 1}{\sqrt{2} - \left( -\sqrt{8} \right)} \end{align*}$. Now you just need to simplify.
$\displaystyle \displaystyle \begin{align*} \frac{3-1}{\sqrt{2} - \left(-\sqrt{8} \right)} &= \frac{2}{\sqrt{2} + \sqrt{8}} \\ &= \frac{2}{\sqrt{2} + 2\sqrt{2}} \\ &= \frac{2}{3\sqrt{2}} \\ &= \frac{2\sqrt{2}}{3\left( \sqrt{2} \right)^2} \\ &= \frac{2\sqrt{2}}{3 \cdot 2} \\ &= \frac{\sqrt{2}}{3} \end{align*}$