Answer
$P(x)=(2x-3)(x+4)$
Work Step by Step
Using the factoring of trinomials in the form $ax^2+bx+c,$ the expression
\begin{align*}
2x^2+5x-12
\end{align*} has $ac=
2(-12)=-24
$ and $b=
5
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
-3,8
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{align*}
2x^2-3x+8x-12
.\end{align*}
Grouping the first and second terms and the third and fourth terms, the expression above is equivalent to
\begin{align*}
(2x^2-3x)+(8x-12)
.\end{align*}
Factoring the $GCF$ in each group results to
\begin{align*}
x(2x-3)+4(2x-3)
.\end{align*}
Factoring the $GCF=
(2x-3)
$ of the entire expression above results to
\begin{align*}
&
(2x-3)(x+4)
.\end{align*}
Hence, the factored form of $
P(x)=2x^2+5x-12
$ is $
P(x)=(2x-3)(x+4)
$.
