#### Answer

$2$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
(\sqrt[3]{3}-1)(\sqrt[3]{9}+\sqrt[3]{3}+1)
,$ use the factoring of 2 cubes.
$\bf{\text{Solution Details:}}$
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}
(\sqrt[3]{3}-1)[(\sqrt[3]{3})^2+\sqrt[3]{3}(1)+(1)^2]
\\\\=
(\sqrt[3]{3})^3-(1)^3
\\\\=
3-1
\\\\=
2
.\end{array}