Answer
$1-5\sqrt[]{15}$
Work Step by Step
Using $(a+b)(c+d)=ac+ad+bc+bd$, or the Distributive Property, and the properties of radicals, the given expression, $
(2\sqrt[]{5}+\sqrt{3})(\sqrt[]{5}-3\sqrt{3})
,$ simplifies to
\begin{array}{l}\require{cancel}
2\sqrt[]{5}(\sqrt[]{5})+2\sqrt[]{5}(-3\sqrt{3})+\sqrt{3}(\sqrt[]{5})+\sqrt{3}(-3\sqrt{3})
\\\\=
2(\sqrt[]{5^{2}})-2(3)\sqrt[]{5(3)}+\sqrt{3(5)}-3(\sqrt{3^{2}})
\\\\=
2(5)-6\sqrt[]{15}+\sqrt{15}-3(3)
\\\\=
10-6\sqrt[]{15}+\sqrt{15}-9
\\\\=
(10-9)+(-6\sqrt[]{15}+\sqrt{15})
\\\\=
1-5\sqrt[]{15}
.\end{array}
