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 $ax^{2}+bx+c=0$.
$$9x^{2}+12x+1=0$$
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 $x=\frac{-12±6\sqrt{3}}{18}$ when $±$ is plus. Add $-12$ to $6\sqrt{3}$.
$$x=\frac{6\sqrt{3}-12}{18}$$
Divide $-12+6\sqrt{3}$ by $18$.
$$x=\frac{\sqrt{3}-2}{3}$$
Now solve the equation $x=\frac{-12±6\sqrt{3}}{18}$ when $±$ is minus. Subtract $6\sqrt{3}$ from $-12$.
$$x=\frac{-6\sqrt{3}-12}{18}$$
Divide $-12-6\sqrt{3}$ by $18$.
$$x=\frac{-\sqrt{3}-2}{3}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{-2+\sqrt{3}}{3}$ for $x_{1}$ and $\frac{-2-\sqrt{3}}{3}$ for $x_{2}$.
