Answer
$\dfrac{20}{\sqrt{10}+\sqrt{15}}$
Work Step by Step
Multiplying by an expression equal to $1$ which will make the numerator a perfect power of the index, then the rationalized-numerator form of the given expression is
\begin{array}{l}\require{cancel}
\dfrac{4\sqrt{5}}{\sqrt{2}+\sqrt{3}}
\\\\=
\dfrac{4\sqrt{5}}{\sqrt{2}+\sqrt{3}}\cdot\dfrac{4\sqrt{5}}{4\sqrt{5}}
\\\\=
\dfrac{(4\sqrt{5})^2}{4\sqrt{5}(\sqrt{2}+\sqrt{3})}
\\\\=
\dfrac{16(5)}{4\sqrt{5}(\sqrt{2}+\sqrt{3})}
\\\\=
\dfrac{80}{4\sqrt{5}(\sqrt{2}+\sqrt{3})}
.\end{array}
Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{80}{4\sqrt{5}(\sqrt{2}+\sqrt{3})}
\\\\=
\dfrac{80}{4\sqrt{5}(\sqrt{2})+4\sqrt{5}(\sqrt{3})}
\\\\=
\dfrac{80}{4\sqrt{10}+4\sqrt{15}}
\\\\=
\dfrac{\cancel4^{20}}{\cancel4\sqrt{10}+\cancel4\sqrt{15}}
\\\\=
\dfrac{20}{\sqrt{10}+\sqrt{15}}
.\end{array}
