Answer
$(2z-1)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
6z^2-11z+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.
$\bf{\text{Solution Details:}}$
In the given expression the value of $ac$ is $
6(4)=24
$ and the value of $b$ is $
-11
.$
The possible pairs of integers whose product is $ac$ are
\begin{array}{l}\require{cancel}
\{1,24\}, \{2,12\}, \{3,8\}, \{4,6\},
\{-1,-24\}, \{-2,-12\}, \{-3,-8\}, \{-4,-6\}
.\end{array}
Among these pairs, the one that gives a sum of $b$ is $\{
-3,-8
\}.$ Using these $2$ numbers to decompose the middle term of the given expression results to
\begin{array}{l}\require{cancel}
6z^2-3z-8z+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}
(6z^2-3z)-(8z-4)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
3z(2z-1)-4(2z-1)
.\end{array}
Factoring the $GCF=
(2z-1)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(2z-1)(3z-4)
.\end{array}
The missing factor of the given expression is $
(2z-1)
.$
