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 $ax^{2}+bx+c=0$.
$$x^{2}-5x-1=0$$
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 $x=\frac{5±\sqrt{29}}{2}$ when $±$ is plus. Add $5$ to $\sqrt{29}$.
$$x=\frac{\sqrt{29}+5}{2}$$
Now solve the equation $x=\frac{5±\sqrt{29}}{2}$ when $±$ is minus. Subtract $\sqrt{29}$ from $5$.
$$x=\frac{5-\sqrt{29}}{2}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{5+\sqrt{29}}{2}$ for $x_{1}$ and $\frac{5-\sqrt{29}}{2}$ for $x_{2}$.
