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 $ax^{2}+bx+c=0$.
$$-x^{2}+3x-1=0$$
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 $x=\frac{-3±\sqrt{5}}{-2}$ when $±$ is plus. Add $-3$ to $\sqrt{5}$.
$$x=\frac{\sqrt{5}-3}{-2}$$
Divide $-3+\sqrt{5}$ by $-2$.
$$x=\frac{3-\sqrt{5}}{2}$$
Now solve the equation $x=\frac{-3±\sqrt{5}}{-2}$ when $±$ is minus. Subtract $\sqrt{5}$ from $-3$.
$$x=\frac{-\sqrt{5}-3}{-2}$$
Divide $-3-\sqrt{5}$ by $-2$.
$$x=\frac{\sqrt{5}+3}{2}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{3-\sqrt{5}}{2}$ for $x_{1}$ and $\frac{3+\sqrt{5}}{2}$ for $x_{2}$.
