$\begin{array}{l}
\dfrac{{\sqrt {21 + 8\sqrt 5 } }}{{4 + \sqrt 5 }}.\sqrt {9 - 4\sqrt 5 } \\
= \dfrac{{\sqrt {16 + 5 + 2.4\sqrt 5 } }}{{4 + \sqrt 5 }}.\sqrt {5 + 4 - 2.2\sqrt 5 } \\
= \dfrac{{\sqrt {{{\left( {4 + \sqrt 5 } \right)}^2}} }}{{4 + \sqrt 5 }}.\sqrt {{{\left( {\sqrt 5 - 2} \right)}^2}} \\
= \dfrac{{4 + \sqrt 5 }}{{4 + \sqrt 5 }}.\left| {\sqrt 5 - 2} \right| = \sqrt 5 - 2
\end{array}$
