Answer
$6a^6 (4a+5b)(2a-3b)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
48a^8-12a^7b-90a^6b^2
,$ 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.
$\bf{\text{Solution Details:}}$
The $GCF$ of the constants of the terms $\{
48,-12,-90
\}$ is $
6
.$ The $GCF$ of the common variable/s is the variable/s with the lowest exponent. Hence, the $GCF$ of the common variable/s $\{
a^8,a^7,a^6
\}$ is $
a^6
.$ Hence, the entire expression has $GCF=
6a^6
.$
Factoring the $GCF=
6a^6
,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
6a^6 \left( 8a^2-2ab-15b^2 \right)
.\end{array}
In the trinomial expression above the value of $ac$ is $
8(-15)=-120
$ and the value of $b$ is $
-2
.$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\{
10,-12
\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
6a^6 \left( 8a^2+10ab-12ab-15b^2 \right)
.\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}
6a^6 [( 8a^2+10ab)-(12ab+15b^2)]
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
6a^6 [2a( 4a+5b)-3b(4a+5b)]
.\end{array}
Factoring the $GCF=
(4a+5b)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
6a^6 [(4a+5b)(2a-3b)]
\\\\=
6a^6 (4a+5b)(2a-3b)
.\end{array}
