Factor 2x^2 - 3x - 2.

Step-by-step solution

1. 1

   Compute the AC product. With $a=2$ and $c=-2$:

   $ac=2\cdot(-2)=-4$

   and we need two integers with product $-4$ and sum $b=-3$.

2. 2

   Find the pair. The candidates for $-4$ are $(1,-4)$, $(2,-2)$, $(4,-1)$ with sums $-3$, $0$, $3$. The winner is $-4$ and $1$.

3. 3

   Split the middle term and group.

   $2x^2-4x+x-2=\left(2x^2-4x\right)+(x-2)$

   $=2x(x-2)+1(x-2)$

   The identical binomial $(x-2)$ in both groups confirms the split was correct — if it had not matched, the pair was wrong or the grouping needs reordering.

4. 4

   Factor out the common binomial.

   $2x^2-3x-2=(x-2)(2x+1)$

5. 5

   Expand back to verify. $(x-2)(2x+1)=2x^2+x-4x-2=2x^2-3x-2$ $\checkmark$. The corresponding roots would be $x=2$ and $x=-\tfrac12$.