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 $ax^{2}+bx+c=0$.
$$5x^{2}+12x+6=0$$
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 $x=\frac{-12±2\sqrt{6}}{10}$ when $±$ is plus. Add $-12$ to $2\sqrt{6}$.
$$x=\frac{2\sqrt{6}-12}{10}$$
Divide $-12+2\sqrt{6}$ by $10$.
$$x=\frac{\sqrt{6}-6}{5}$$
Now solve the equation $x=\frac{-12±2\sqrt{6}}{10}$ when $±$ is minus. Subtract $2\sqrt{6}$ from $-12$.
$$x=\frac{-2\sqrt{6}-12}{10}$$
Divide $-12-2\sqrt{6}$ by $10$.
$$x=\frac{-\sqrt{6}-6}{5}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{-6+\sqrt{6}}{5}$ for $x_{1}$ and $\frac{-6-\sqrt{6}}{5}$ for $x_{2}$.
