Using synthetic division a number of times, you find factorization

At this point, I would recommend using the quadratic formula to get the zeroes of the factor $2x^2 - 1$. Of course, this polynomial is the same as $2x^2 + 0x - 1$. Finding the roots of that polynomial is the same as finding the solutions of the equation $2x^2 + 0x - 1 = 0$. That question can be answered by using the quadratic formula: $$\begin{align*} 0 &= 2x^2 + 0x - 1 \\ x &= \frac{-0 \pm \sqrt{0^2 - 4\cdot 2\cdot (-1)}}{2(2)} \\ x &= \frac{\pm \sqrt{8}}{4} \\ x &= \frac{\pm \sqrt{4\cdot 2}}{4} \\ x &= \frac{\pm\sqrt{4}\sqrt{2}}{4} \\ x&= \pm \frac{2\sqrt{2}}{4} \\ x &= \pm \frac{\sqrt{2}}{2} \\ x &= \pm \frac{1}{\sqrt{2}} \qquad \text{(``rationalizing the denominator'')}\end{align*}$$