Answer
$(8p-3)(3p-4)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
24p^2-41p+12
,$ find two numbers whose product is $ac$ and whose sum is $b$ in the quadratic expression $ax^2+bx+c.$ Use these $2$ numbers to decompose the middle term of the given quadratic expression and then use factoring by grouping.
$\bf{\text{Solution Details:}}$
Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $
24(12)=288
$ and the value of $b$ is $
-41
.$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{
-9,-32
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
24p^2-9p-32p+12
.\end{array}
Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to
\begin{array}{l}\require{cancel}
(24p^2-9p)-(32p-12)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
3p(8p-3)-4(8p-3)
.\end{array}
Factoring the $GCF=
(8p-3)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(8p-3)(3p-4)
.\end{array}
