Answer
$8\sqrt[3]{21}+20\sqrt[3]{18}+2\sqrt[3]{70}+5\sqrt[3]{60}$
Work Step by Step
Using $(a+b)(c+d)=ac+ad+bc+bd$, or the product of 2 binomials, and the properties of radicals, the given expression, $
(4\sqrt[3]{3}+\sqrt[3]{10})(2\sqrt[3]{7}+5\sqrt[3]{6})
,$ is equivalent to
\begin{array}{l}\require{cancel}
(4\sqrt[3]{3})(2\sqrt[3]{7})+(4\sqrt[3]{3})(5\sqrt[3]{6})+(\sqrt[3]{10})(2\sqrt[3]{7})+(\sqrt[3]{10})(5\sqrt[3]{6})
\\\\=
4(2)\sqrt[3]{3(7)}+4(5)\sqrt[3]{3(6)}+2\sqrt[3]{10(7)}+5\sqrt[3]{10(6)}
\\\\=
8\sqrt[3]{21}+20\sqrt[3]{18}+2\sqrt[3]{70}+5\sqrt[3]{60}
.\end{array}
