Answer
$-6$
Work Step by Step
Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
\sqrt[3]{2}(\sqrt[3]{4}-2\sqrt[3]{32})
\\\\=
\sqrt[3]{2}(\sqrt[3]{4})+\sqrt[3]{2}(-2\sqrt[3]{32})
.\end{array}
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\sqrt[3]{2}(\sqrt[3]{4})+\sqrt[3]{2}(-2\sqrt[3]{32})
\\\\=
\sqrt[3]{2(4)}-2\sqrt[3]{2(32)}
\\\\=
\sqrt[3]{8}-2\sqrt[3]{64}
.\end{array}
Extracting the root of the factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
\sqrt[3]{8}-2\sqrt[3]{64}
\\\\=
\sqrt[3]{(2)^3}-2\sqrt[3]{(4)^3}
\\\\=
2-2(4)
\\\\=
2-8
\\\\=
-6
.\end{array}
