Multiply:
$ \left(3 x^{2} y-5 x y^{2}\right) $ by $ \left(\frac{1}{5} x^{2}+\frac{1}{3} y^{2}\right) $

To do:

We have to multiply \( \left(3 x^{2} y-5 x y^{2}\right) \) by \( \left(\frac{1}{5} x^{2}+\frac{1}{3} y^{2}\right) \)

Solution:

We know that,

$(a+b)\times(c+d)=a(c+d)+b(c+d)$

Therefore,

$(3 x^{2} y-5 x y^{2}) \times(\frac{1}{5} x^{2}+\frac{1}{3} y^{2})=3 x^{2} y \times(\frac{1}{5} x^{2}+\frac{1}{3} y^{2})-5 x y^{2}\times(\frac{1}{5} x^{2}+\frac{1}{3} y^{2})$

$=3 x^{2} y \times \frac{1}{5} x^{2}+3 x^{2} y \times \frac{1}{3} y^{2}-5 x y^{2} \times \frac{1}{5} x^{2}-5 x y^{2} \times \frac{1}{3} y^{2}$

$=\frac{3}{5} x^{2+2} y+x^{2} y^{1+2}-x^{1+2} y^{2}-\frac{5}{3} xy^{2+2}$

$=\frac{3}{5} x^{4} y+x^{2} y^{3}-x^{3} y^{2}-\frac{5}{3} x y^{4}$

$=\frac{3}{5} x^{4} y-x^{3} y^{2}+x^{2} y^{3}-\frac{5}{3} x y^{4}$