Factor the expression by grouping. First, the expression needs to be rewritten as $A^{2}+aA+bA-2$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-1$$ $$ab=1\left(-2\right)=-2$$
Since $ab$ is negative, $a$ and $b$ have the opposite signs. Since $a+b$ is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.
$$a=-2$$ $$b=1$$
Rewrite $A^{2}-A-2$ as $\left(A^{2}-2A\right)+\left(A-2\right)$.
$$\left(A^{2}-2A\right)+\left(A-2\right)$$
Factor out $A$ in $A^{2}-2A$.
$$A\left(A-2\right)+A-2$$
Factor out common term $A-2$ by using distributive property.
$$\left(A-2\right)\left(A+1\right)$$
Steps Using the Quadratic Formula
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 $ax^{2}+bx+c=0$.
$$A^{2}-A-2=0$$
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.
