Answer
$75+34\sqrt{5}$
Work Step by Step
Using the FOIL Method which is given by $(a+b)(c+d)=ac+ad+bc+bd,$ the given expression, $
(5+2\sqrt{5})(7+4\sqrt{5})
,$ is equivalent to
\begin{align*}
&
5(7)+5(4\sqrt{5})+2\sqrt{5}(7)+2\sqrt{5}(4\sqrt{5})
\\&=
35+20\sqrt{5}+14\sqrt{5}+8(5)
\\&=
35+20\sqrt{5}+14\sqrt{5}+40
\\&=
(35+40)+(20\sqrt{5}+14\sqrt{5})
\\&=
75+34\sqrt{5}
.\end{align*}
Hence, the product of the given expression is $
75+34\sqrt{5}
$.
