Factoring Trinomials of the Form x² + bx+ c, Where c is Negative

$$\begin{align} &x^2 - x - 2\cr &\quad = x^2 + (\overset{=\ -1}{\overbrace{(-2)+1}})x + \overset{=\ -2}{\overbrace{\ (-2)\cdot 1\ }}\cr \cr&\quad = (x - 2)(x + 1)\end{align}$$

$$x^2+x−9 $$ is not factorable over the integers

$$x^2−2x−12$$ is not factorable over the integers

$$\begin{align} &x^2 + 2x - 3\cr &\quad = x^2 + (\overset{=\ 2}{\overbrace{3+(-1)}})x + \overset{=\ -3}{\overbrace{\ 3\cdot (-1)\ }}\cr \cr &\quad = (x + 3)(x - 1)\end{align}$$

$$\begin{align} &x^2 - 6x - 27\cr &\quad = x^2 + (\overset{=\ -6}{\overbrace{(-9)+3}})x + \overset{=\ -27}{\overbrace{\ (-9)\cdot 3\ }}\cr \cr&\quad = (x - 9)(x + 3)\end{align}$$

$$x^2+x−8$$ is not factorable over the integers

$$\begin{align} &x^2 - 6x - 7\cr &\quad = x^2 + (\overset{=\ -6}{\overbrace{(-7)+1}})x + \overset{=\ -7}{\overbrace{\ (-7)\cdot 1\ }}\cr \cr&\quad = (x - 7)(x + 1)\end{align}$$