Multiply:

To multiply, we will use distributive law as follows:

$\left(\frac{3}{5} x+\frac{1}{2} y\right)\left(\frac{5}{6} x+4 y\right)$

$=\frac{3}{5} x\left(\frac{5}{6} x+4 y\right)+\frac{1}{2} y\left(\frac{5}{6} x+4 y\right)$

$=\frac{1}{2} x^{2}+\frac{12}{5} x y+\frac{5}{12} x y+2 y^{2}$

$=\frac{1}{2} x^{2}+\left(\frac{144+25}{60}\right) x y+2 y^{2}$

$=\frac{1}{2} x^{2}+\frac{169}{60} x y+2 y^{2}$

Thus, the answer is $\frac{1}{2} x^{2}+\frac{169}{60} x y+2 y^{2}$.