#### Answer

$\dfrac{20-4\sqrt{6}}{19}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To rationalize the given radical expression, $
\dfrac{4}{5+\sqrt{6}}
,$ multiply the numerator and the denominator by the conjugate of the denominator. Then use special products to multiply the result.
$\bf{\text{Solution Details:}}$
Multiplying the numerator and the denominator of the given expression by the conjugate of the denominator, the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{4}{5+\sqrt{6}}\cdot\dfrac{5-\sqrt{6}}{5-\sqrt{6}}
\\\\=
\dfrac{4(5-\sqrt{6})}{(5+\sqrt{6})(5-\sqrt{6})}
.\end{array}
Using the product of the sum and difference of like terms which is given by $(a+b)(a-b)=a^2-b^2,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
\dfrac{4(5-\sqrt{6})}{(5)^2-(\sqrt{6})^2}
\\\\=
\dfrac{4(5-\sqrt{6})}{25-6}
\\\\=
\dfrac{4(5-\sqrt{6})}{19}
.\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{4(5)-4(\sqrt{6})}{19}
\\\\=
\dfrac{20-4\sqrt{6}}{19}
.\end{array}