Answer
$2(3m-10)(9m^2+30m+100)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
54m^3-2000
,$ factor first the $GCF.$ Then use the factoring of the sum or difference of $2$ cubes.
$\bf{\text{Solution Details:}}$
The $GCF$ of the terms in the given expression is $
2
,$ since it is the greatest expression that can divide all the terms evenly (no remainder.) Factoring the $GCF$ results to
\begin{array}{l}\require{cancel}
2(27m^3-1000)
.\end{array}
The expressions $
27m^3
$ and $
1000
$ are both perfect cubes (the cube root is exact). Hence, $
27m^3-1000
$ is a $\text{
difference
}$ of $2$ cubes. Using the factoring of the sum or difference of $2$ cubes which is given by $a^3+b^3=(a+b)(a^2-ab+b^2)$ or by $a^3-b^3=(a-b)(a^2+ab+b^2)$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
2[(3m)^3-(10)^3]
\\\\=
2(3m-10)[(3m)^2+(3m)(10)+(10^2]
\\\\=
2(3m-10)(9m^2+30m+100)
.\end{array}
