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 $ax^{2}+bx+c=0$.
$$a^{2}+7a+9=0$$
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.
$$a=\frac{-7±\sqrt{7^{2}-4\times 9}}{2}$$
Square $7$.
$$a=\frac{-7±\sqrt{49-4\times 9}}{2}$$
Multiply $-4$ times $9$.
$$a=\frac{-7±\sqrt{49-36}}{2}$$
Add $49$ to $-36$.
$$a=\frac{-7±\sqrt{13}}{2}$$
Now solve the equation $a=\frac{-7±\sqrt{13}}{2}$ when $±$ is plus. Add $-7$ to $\sqrt{13}$.
$$a=\frac{\sqrt{13}-7}{2}$$
Now solve the equation $a=\frac{-7±\sqrt{13}}{2}$ when $±$ is minus. Subtract $\sqrt{13}$ from $-7$.
$$a=\frac{-\sqrt{13}-7}{2}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{-7+\sqrt{13}}{2}$ for $x_{1}$ and $\frac{-7-\sqrt{13}}{2}$ for $x_{2}$.
