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 $ax^{2}+bx+c=0$.
$$3x^{2}+3x-3=0$$
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 $x=\frac{-3±3\sqrt{5}}{6}$ when $±$ is plus. Add $-3$ to $3\sqrt{5}$.
$$x=\frac{3\sqrt{5}-3}{6}$$
Divide $-3+3\sqrt{5}$ by $6$.
$$x=\frac{\sqrt{5}-1}{2}$$
Now solve the equation $x=\frac{-3±3\sqrt{5}}{6}$ when $±$ is minus. Subtract $3\sqrt{5}$ from $-3$.
$$x=\frac{-3\sqrt{5}-3}{6}$$
Divide $-3-3\sqrt{5}$ by $6$.
$$x=\frac{-\sqrt{5}-1}{2}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{-1+\sqrt{5}}{2}$ for $x_{1}$ and $\frac{-1-\sqrt{5}}{2}$ for $x_{2}$.
