Quadratic polynomial can be factored using the transformation $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$, where $x_{1}$ and $x_{2}$ are the solutions of the quadratic equation $ax^{2}+bx+c=0$.
$$2a^{2}-a-2=0$$
All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. The quadratic formula gives two solutions, one when $±$ is addition and one when it is subtraction.
Now solve the equation $a=\frac{1±\sqrt{17}}{4}$ when $±$ is plus. Add $1$ to $\sqrt{17}$.
$$a=\frac{\sqrt{17}+1}{4}$$
Now solve the equation $a=\frac{1±\sqrt{17}}{4}$ when $±$ is minus. Subtract $\sqrt{17}$ from $1$.
$$a=\frac{1-\sqrt{17}}{4}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{1+\sqrt{17}}{4}$ for $x_{1}$ and $\frac{1-\sqrt{17}}{4}$ for $x_{2}$.
