Solve x+2y+4y^2/x-2y-4x^2y/x^2-4y^2

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

Evaluate

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

Short Solution Steps

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

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

Since \frac{\left(x+2y\right)\left(x-2y\right)}{x-2y} and \frac{4y^{2}}{x-2y} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x+2y\right)\left(x-2y\right)+4y^{2}.

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

Combine like terms in x^{2}-2xy+2yx-4y^{2}+4y^{2}.

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

Factor x^{2}-4y^{2}.

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

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

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

Since \frac{x^{2}\left(x+2y\right)}{\left(x-2y\right)\left(x+2y\right)} and \frac{4x^{2}y}{\left(x-2y\right)\left(x+2y\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x^{2}\left(x+2y\right)-4x^{2}y.

$$\frac{x^{3}+2x^{2}y-4x^{2}y}{\left(x-2y\right)\left(x+2y\right)}$$

Combine like terms in x^{3}+2x^{2}y-4x^{2}y.

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

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

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

Cancel out x-2y in both numerator and denominator.

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

Differentiate w.r.t. x

$$\frac{x\left(x+4y\right)}{\left(x+2y\right)^{2}}$$