#### Answer

$(p+1)(3p-4)$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To factor the given expression, $
3p^2-p-4
,$ 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.
$\bf{\text{Solution Details:}}$
To factor the trinomial expression above, note that the value of $ac$ is $
3(-4)=-12
$ and the value of $b$ is $
-1
.$
The possible pairs of integers whose product is $ac$ are
\begin{array}{l}\require{cancel}
\{1,-12\}, \{2,-6\}, \{3,-4\},
\\
\{-1,12\}, \{-2,6\}, \{-3,4\}
.\end{array}
Among these pairs, the one that gives a sum of $b$ is $\{
3,-4
\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
3p^2+3p-4p-4
.\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}
(3p^2+3p)-(4p+4)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
3p(p+1)-4(p+1)
.\end{array}
Factoring the $GCF=
(p+1)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(p+1)(3p-4)
.\end{array}