Answer
$(2x-3)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
4x^2-4x-3
,$ 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.
$\bf{\text{Solution Details:}}$
In the given expression the value of $ac$ is $
4(-3)=-12
$ and the value of $b$ is $
-4
.$
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 $\{
2,-6
\}.$ Using these $2$ numbers to decompose the middle term of the given expression results to
\begin{array}{l}\require{cancel}
4x^2+2x-6x-3
.\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}
(4x^2+2x)-(6x+3)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
2x(2x+1)-3(2x+1)
.\end{array}
Factoring the $GCF=
(2x+1)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(2x+1)(2x-3)
.\end{array}
The missing factor of the given expression is $
(2x-3)
.$
