Answer
$(7c+2d)(2c-3d)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the quadratic expression $ax^2+bx+c,$ find two numbers whose product is $ac$ and whose sum is $b$. Use these $2$ numbers to decompose the middle term of the quadratic expression and then use factoring by grouping.
$\bf{\text{Solution Details:}}$
In the given expression, $
14c^2-17cd-6d^2
,$ the value of $ac$ is $
14(-6)=-84
$ and the value of $b$ is $
-17
.$
The possible pairs of integers whose product is $ac$ are
\begin{array}{l}\require{cancel}
\{1,-84\}, \{2,-42\}, \{3,-28\}, \{4,-21\}, \{7,-12\},
\{-1,84\}, \{-2,42\}, \{-3,28\}, \{-4,21\}, \{-7,12\}
.\end{array}
Among these pairs, the one that gives a sum of $b$ is $\{
4,-21
\}.$ Using these $2$ numbers to decompose the middle term of the given expression results to
\begin{array}{l}\require{cancel}
14c^2+4cd-21cd-6d^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}
(14c^2+4cd)-(21cd+6d^2)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
2c(7c+2d)-3d(7c+2d)
.\end{array}
Factoring the $GCF=
(7c+2d)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(7c+2d)(2c-3d)
.\end{array}
