Answer
$2b(3a+7)^2$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
18a^2b+84ab+98b
,$ factor first the $GCF.$ Then 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.
$\bf{\text{Solution Details:}}$
Factoring the $GCF=
2b
,$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
2b(9a^2+42a+49)
.\end{array}
Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $
9(49)=441
$ and the value of $b$ is $
42
.$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{
21,21
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
2b(9a^2+21a+21a+49)
.\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}
2b[(9a^2+21a)+(21a+49)]
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
2b[3a(3a+7)+7(3a+7)]
.\end{array}
Factoring the $GCF=
(3a+7)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
2b[(3a+7)(3a+7)]
\\\\=
2b(3a+7)^2
.\end{array}