# How do you simplify (x-y)/(x+y)*(x^2+2xy+y^2)/(x^2-2xy+y^2) div (x^2-y^2)/( x^2-2xy+3y^2)?

Jul 21, 2015

Resolve into linear factors, cancel factors common to numerator and denominator and simplify to find:

$\frac{x - y}{x + y} \cdot \frac{{x}^{2} + 2 x y + {y}^{2}}{{x}^{2} - 2 x y + {y}^{2}} \div \frac{{x}^{2} - {y}^{2}}{{x}^{2} - 2 x y + 3 {y}^{2}}$

$= 1 - \frac{4 {y}^{2}}{{\left(x - y\right)}^{2}}$

#### Explanation:

$\frac{x - y}{x + y} \cdot \frac{{x}^{2} + 2 x y + {y}^{2}}{{x}^{2} - 2 x y + {y}^{2}} \div \frac{{x}^{2} - {y}^{2}}{{x}^{2} - 2 x y + 3 {y}^{2}}$

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

$= \frac{\left(x - y\right) \left({x}^{2} + 2 x y + {y}^{2}\right) \left({x}^{2} - 2 x y + 3 {y}^{2}\right)}{\left(x + y\right) \left({x}^{2} - 2 x y + {y}^{2}\right) \left({x}^{2} - {y}^{2}\right)}$

$= \frac{\left(x - y\right) \left(x + y\right) \left(x + y\right) \left(x + y\right) \left(x - 3 y\right)}{\left(x + y\right) \left(x - y\right) \left(x - y\right) \left(x - y\right) \left(x + y\right)}$

$= \frac{\left(x + y\right) \left(x - 3 y\right)}{\left(x - y\right) \left(x - y\right)}$

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

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

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

$= 1 - \frac{4 {y}^{2}}{{\left(x - y\right)}^{2}}$