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 $ax^{2}+bx+c=0$.
$$5x^{2}-2x-2=0$$
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 $x=\frac{2±2\sqrt{11}}{10}$ when $±$ is plus. Add $2$ to $2\sqrt{11}$.
$$x=\frac{2\sqrt{11}+2}{10}$$
Divide $2+2\sqrt{11}$ by $10$.
$$x=\frac{\sqrt{11}+1}{5}$$
Now solve the equation $x=\frac{2±2\sqrt{11}}{10}$ when $±$ is minus. Subtract $2\sqrt{11}$ from $2$.
$$x=\frac{2-2\sqrt{11}}{10}$$
Divide $2-2\sqrt{11}$ by $10$.
$$x=\frac{1-\sqrt{11}}{5}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{1+\sqrt{11}}{5}$ for $x_{1}$ and $\frac{1-\sqrt{11}}{5}$ for $x_{2}$.
