Answer
$5r^2-\dfrac{73}{3}ry-\dfrac{10}{3}y^2$
Work Step by Step
Using $(a+b)(c+d)=ac+ad+bc+bd$ or the FOIL Method, the product of the binomials in the given expression, $
\left( 5r+\dfrac{2}{3}y \right)( r-5y )
,$ is
\begin{array}{l}\require{cancel}
5r(r)+5r(-5y )+\dfrac{2}{3}y(r)+\dfrac{2}{3}y(-5y)
\\\\=
5r^2-25ry+\dfrac{2}{3}ry-\dfrac{10}{3}y^2
\\\\=
5r^2-\dfrac{75}{3}ry+\dfrac{2}{3}ry-\dfrac{10}{3}y^2
\\\\=
5r^2-\dfrac{73}{3}ry-\dfrac{10}{3}y^2
.\end{array}