Solve frac{x^{2}-9y^{2}}{x^{2}+5xy+6y^{2}}divx^2-2xy-3y^2/x^2-y^2

$$\frac { x ^ { 2 } - 9 y ^ { 2 } } { x ^ { 2 } + 5 x y + 6 y ^ { 2 } } \div \frac { x ^ { 2 } - 2 x y - 3 y ^ { 2 } } { x ^ { 2 } - y ^ { 2 } }$$

Evaluate

$$\frac{x-y}{x+2y}$$

Short Solution Steps

Divide \frac{x^{2}-9y^{2}}{x^{2}+5xy+6y^{2}} by \frac{x^{2}-2xy-3y^{2}}{x^{2}-y^{2}} by multiplying \frac{x^{2}-9y^{2}}{x^{2}+5xy+6y^{2}} by the reciprocal of \frac{x^{2}-2xy-3y^{2}}{x^{2}-y^{2}}.

$$\frac{\left(x^{2}-9y^{2}\right)\left(x^{2}-y^{2}\right)}{\left(x^{2}+5xy+6y^{2}\right)\left(x^{2}-2xy-3y^{2}\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x+y\right)\left(x-3y\right)\left(x+3y\right) in both numerator and denominator.

$$\frac{x-y}{x+2y}$$

Expand

$$\frac{x-y}{x+2y}$$

Short Solution Steps

Divide \frac{x^{2}-9y^{2}}{x^{2}+5xy+6y^{2}} by \frac{x^{2}-2xy-3y^{2}}{x^{2}-y^{2}} by multiplying \frac{x^{2}-9y^{2}}{x^{2}+5xy+6y^{2}} by the reciprocal of \frac{x^{2}-2xy-3y^{2}}{x^{2}-y^{2}}.

$$\frac{\left(x^{2}-9y^{2}\right)\left(x^{2}-y^{2}\right)}{\left(x^{2}+5xy+6y^{2}\right)\left(x^{2}-2xy-3y^{2}\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x+y\right)\left(x-3y\right)\left(x+3y\right) in both numerator and denominator.

$$\frac{x-y}{x+2y}$$