jmc

The given expression is equivalent to the product $$\left(\sqrt[3]{4}+\sqrt[3]{2}\right)\cdot \left(-\sqrt[3]{16}+\sqrt[3]{8}-\sqrt[3]{4}\right).$$If we let $a=\sqrt[3]{4}$ and $b=\sqrt[3]{2}$, then the above expression becomes $$(a+b)(-a^2+ab-b^2)=-(a+b)(a^2-ab+b^2)=-(a^3+b^3).$$Thus, the expression is equal to $-{\left(\sqrt[3]{4}\right)}^3-{\left(\sqrt[3]{2}\right)}^3=\boxed{-6}$.