Solve x(x+y)/(x-y)^2*2x/x+y-1

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

Evaluate

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

Short Solution Steps

Use the distributive property to multiply x by x+y.

$$\frac{x^{2}+xy}{\left(x-y\right)^{2}}\times \frac{2x}{x+y}-1$$

Multiply \frac{x^{2}+xy}{\left(x-y\right)^{2}} times \frac{2x}{x+y} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x^{2}+xy\right)\times 2x}{\left(x-y\right)^{2}\left(x+y\right)}-1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(x-y\right)^{2}\left(x+y\right)}{\left(x-y\right)^{2}\left(x+y\right)}.

$$\frac{\left(x^{2}+xy\right)\times 2x}{\left(x-y\right)^{2}\left(x+y\right)}-\frac{\left(x-y\right)^{2}\left(x+y\right)}{\left(x-y\right)^{2}\left(x+y\right)}$$

Since \frac{\left(x^{2}+xy\right)\times 2x}{\left(x-y\right)^{2}\left(x+y\right)} and \frac{\left(x-y\right)^{2}\left(x+y\right)}{\left(x-y\right)^{2}\left(x+y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{2}+xy\right)\times 2x-\left(x-y\right)^{2}\left(x+y\right)}{\left(x-y\right)^{2}\left(x+y\right)}$$

Do the multiplications in \left(x^{2}+xy\right)\times 2x-\left(x-y\right)^{2}\left(x+y\right).

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

Combine like terms in 2x^{3}+2x^{2}y-x^{3}-x^{2}y+2x^{2}y+2xy^{2}-y^{2}x-y^{3}.

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

Factor the expressions that are not already factored in \frac{x^{3}+xy^{2}+3x^{2}y-y^{3}}{\left(x-y\right)^{2}\left(x+y\right)}.

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

Cancel out x+y in both numerator and denominator.

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

Expand \left(x-y\right)^{2}.

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

Expand

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

Short Solution Steps

Use the distributive property to multiply x by x+y.

$$\frac{x^{2}+xy}{\left(x-y\right)^{2}}\times \frac{2x}{x+y}-1$$

Multiply \frac{x^{2}+xy}{\left(x-y\right)^{2}} times \frac{2x}{x+y} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x^{2}+xy\right)\times 2x}{\left(x-y\right)^{2}\left(x+y\right)}-1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(x-y\right)^{2}\left(x+y\right)}{\left(x-y\right)^{2}\left(x+y\right)}.

$$\frac{\left(x^{2}+xy\right)\times 2x}{\left(x-y\right)^{2}\left(x+y\right)}-\frac{\left(x-y\right)^{2}\left(x+y\right)}{\left(x-y\right)^{2}\left(x+y\right)}$$

Since \frac{\left(x^{2}+xy\right)\times 2x}{\left(x-y\right)^{2}\left(x+y\right)} and \frac{\left(x-y\right)^{2}\left(x+y\right)}{\left(x-y\right)^{2}\left(x+y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{2}+xy\right)\times 2x-\left(x-y\right)^{2}\left(x+y\right)}{\left(x-y\right)^{2}\left(x+y\right)}$$

Do the multiplications in \left(x^{2}+xy\right)\times 2x-\left(x-y\right)^{2}\left(x+y\right).

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

Combine like terms in 2x^{3}+2x^{2}y-x^{3}-x^{2}y+2x^{2}y+2xy^{2}-y^{2}x-y^{3}.

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

Factor the expressions that are not already factored in \frac{x^{3}+xy^{2}+3x^{2}y-y^{3}}{\left(x-y\right)^{2}\left(x+y\right)}.

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

Cancel out x+y in both numerator and denominator.

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

Expand \left(x-y\right)^{2}.

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