#### Answer

$(4p-q)(p+q)$

#### Work Step by Step

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