Solve (x+6/x+2-4/x)divx^2-16/x^2-4

$$( \frac { x + 6 } { x + 2 } - \frac { 4 } { x } ) \div \frac { x ^ { 2 } - 16 } { x ^ { 2 } - 4 }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and x is x\left(x+2\right). Multiply \frac{x+6}{x+2} times \frac{x}{x}. Multiply \frac{4}{x} times \frac{x+2}{x+2}.

$$\frac{\frac{\left(x+6\right)x}{x\left(x+2\right)}-\frac{4\left(x+2\right)}{x\left(x+2\right)}}{\frac{x^{2}-16}{x^{2}-4}}$$

Since \frac{\left(x+6\right)x}{x\left(x+2\right)} and \frac{4\left(x+2\right)}{x\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(x+6\right)x-4\left(x+2\right)}{x\left(x+2\right)}}{\frac{x^{2}-16}{x^{2}-4}}$$

Do the multiplications in \left(x+6\right)x-4\left(x+2\right).

$$\frac{\frac{x^{2}+6x-4x-8}{x\left(x+2\right)}}{\frac{x^{2}-16}{x^{2}-4}}$$

Combine like terms in x^{2}+6x-4x-8.

$$\frac{\frac{x^{2}+2x-8}{x\left(x+2\right)}}{\frac{x^{2}-16}{x^{2}-4}}$$

Divide \frac{x^{2}+2x-8}{x\left(x+2\right)} by \frac{x^{2}-16}{x^{2}-4} by multiplying \frac{x^{2}+2x-8}{x\left(x+2\right)} by the reciprocal of \frac{x^{2}-16}{x^{2}-4}.

$$\frac{\left(x^{2}+2x-8\right)\left(x^{2}-4\right)}{x\left(x+2\right)\left(x^{2}-16\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x+2\right)\left(x+4\right) in both numerator and denominator.

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

Expand the expression.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and x is x\left(x+2\right). Multiply \frac{x+6}{x+2} times \frac{x}{x}. Multiply \frac{4}{x} times \frac{x+2}{x+2}.

$$\frac{\frac{\left(x+6\right)x}{x\left(x+2\right)}-\frac{4\left(x+2\right)}{x\left(x+2\right)}}{\frac{x^{2}-16}{x^{2}-4}}$$

Since \frac{\left(x+6\right)x}{x\left(x+2\right)} and \frac{4\left(x+2\right)}{x\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(x+6\right)x-4\left(x+2\right)}{x\left(x+2\right)}}{\frac{x^{2}-16}{x^{2}-4}}$$

Do the multiplications in \left(x+6\right)x-4\left(x+2\right).

$$\frac{\frac{x^{2}+6x-4x-8}{x\left(x+2\right)}}{\frac{x^{2}-16}{x^{2}-4}}$$

Combine like terms in x^{2}+6x-4x-8.

$$\frac{\frac{x^{2}+2x-8}{x\left(x+2\right)}}{\frac{x^{2}-16}{x^{2}-4}}$$

Divide \frac{x^{2}+2x-8}{x\left(x+2\right)} by \frac{x^{2}-16}{x^{2}-4} by multiplying \frac{x^{2}+2x-8}{x\left(x+2\right)} by the reciprocal of \frac{x^{2}-16}{x^{2}-4}.

$$\frac{\left(x^{2}+2x-8\right)\left(x^{2}-4\right)}{x\left(x+2\right)\left(x^{2}-16\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x+2\right)\left(x+4\right) in both numerator and denominator.

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

Expand the expression.

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