Solve x-2/x+4-2x/x+2

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

Evaluate

$$-\frac{x^{2}+8x+4}{\left(x+2\right)\left(x+4\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+4 and x+2 is \left(x+2\right)\left(x+4\right). Multiply \frac{x-2}{x+4} times \frac{x+2}{x+2}. Multiply \frac{2x}{x+2} times \frac{x+4}{x+4}.

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

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

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

Do the multiplications in \left(x-2\right)\left(x+2\right)-2x\left(x+4\right).

$$\frac{x^{2}+2x-2x-4-2x^{2}-8x}{\left(x+2\right)\left(x+4\right)}$$

Combine like terms in x^{2}+2x-2x-4-2x^{2}-8x.

$$\frac{-x^{2}-8x-4}{\left(x+2\right)\left(x+4\right)}$$

Expand \left(x+2\right)\left(x+4\right).

$$\frac{-x^{2}-8x-4}{x^{2}+6x+8}$$

Expand

$$-\frac{x^{2}+8x+4}{\left(x+2\right)\left(x+4\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+4 and x+2 is \left(x+2\right)\left(x+4\right). Multiply \frac{x-2}{x+4} times \frac{x+2}{x+2}. Multiply \frac{2x}{x+2} times \frac{x+4}{x+4}.

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

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

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

Do the multiplications in \left(x-2\right)\left(x+2\right)-2x\left(x+4\right).

$$\frac{x^{2}+2x-2x-4-2x^{2}-8x}{\left(x+2\right)\left(x+4\right)}$$

Combine like terms in x^{2}+2x-2x-4-2x^{2}-8x.

$$\frac{-x^{2}-8x-4}{\left(x+2\right)\left(x+4\right)}$$

Expand \left(x+2\right)\left(x+4\right).

$$\frac{-x^{2}-8x-4}{x^{2}+6x+8}$$