Answer
$-9k(k-3)(k+7)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
First, factor the GCF of the terms. Then, to factor the quadratic expression $x^2+bx+c,$ find two numbers, $m_1$ and $m_2,$ whose product is $c$ and whose sum is $b$. Express the factored form as $GCF(x+m_1)(x+m_2).$
$\bf{\text{Solution Details:}}$
Using the negative $GCF=
-9k
,$ the given expression, $
-9k^3-36k^2+189k
,$ is equivalent to
\begin{array}{l}\require{cancel}
-9k(k^2+4k-21)
.\end{array}
In the expression above, the value of $c$ is $
-21
$ and the value of $b$ is $
4
.$
The possible pairs of integers whose product is $ac$ are
\begin{array}{l}\require{cancel}
\{1,-21\}, \{3,-7\},
\{-1,21\}, \{-3,7\}
.\end{array}
Among these pairs, the one that gives a sum of $b$ is $\{
-3,7
\}.$ Hence, the factored form of the given expression is
\begin{array}{l}\require{cancel}
-9k(k-3)(k+7)
.\end{array}