Answer
$\frac{3x^2+16xy-13y^2}{(x+5y)(x-5y)(x-4y)}$.
Work Step by Step
The given expression is
$=\frac{3x-y}{x^2-9xy+20y^2}+\frac{2y}{x^2-25y^2}$
First denominator $=x^2-9xy+20y^2$.
Rewrite the middle term $-9xy$ as $-5xy-4xy$
$=x^2-5xy-4xy+20y^2$
Group terms.
$=(x^2-5xy)+(-4xy+20y^2)$
Factor each term.
$=x(x-5y)-4y(x-5y)$
Factor out $(x-5y)$.
$=(x-5y)(x-4y)$.
Second denominator $=x^2-25y^2$.
Use the algebraic identity $a^2-b^2=(a+b)(a-b)$.
$=x^2-(5y)^2$
$=(x+5y)(x-5y)$
Substitute all factors into the given expression.
$=\frac{3x-y}{(x-5y)(x-4y)}+\frac{2y}{(x+5y)(x-5y)}$
The LCM of all the denominators is $=(x+5y)(x-5y)(x-4y)$.
$=\frac{3x-y}{(x-5y)(x-4y)}\times \frac{x+5y}{x+5y}+\frac{2y}{(x+5y)(x-5y)}\times \frac{x-4y}{x-4y}$
$=\frac{(3x-y)(x+5y)}{(x+5y)(x-5y)(x-4y)}+\frac{2y(x-4y)}{(x+5y)(x-5y)(x-4y)}$
$=\frac{(3x-y)(x+5y)+2y(x-4y)}{(x+5y)(x-5y)(x-4y)}$
$=\frac{3x^2+15xy-xy-5y^2+2xy-8y^2}{(x+5y)(x-5y)(x-4y)}$
Simplify.
$=\frac{3x^2+16xy-13y^2}{(x+5y)(x-5y)(x-4y)}$.
