Answer
$\displaystyle \frac{x-y}{2x-y} $
Work Step by Step
Simplifying Rational Expressions
1. Factor the numerator and the denominator completely.
2. Divide both the numerator and the denominator by any common factors.
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Numerator $:$ search for integer factors of $-3$ whose sum is +2...
... these are $3$ and $-1...$
$x^{2}+2xy-3y^{2} =(x+3y)(x-y)$
Denominator:
search for integer factors of $2(-3)=-6$ whose sum is $+5$...
... these are $6$ and $-1...$
$2x^{2} +5xy-3y^{2}=2x^{2} +6xy-xy-3y^{2}$
$=2x(x+3y)-y(x+3y)$
$=(x+3y)(2x-y)$
Expression = $\displaystyle \frac{(x+3y)(x-y)}{(x+3y)(2x-y)}$
... divide both with the common factor: $ (x+3y)$
Expression = $\displaystyle \frac{x-y}{2x-y} $
