Solve (2)x^2-8x+16/x^2+2xdiv(12/x+2-x+2)

$$( 2 ) \frac { x ^ { 2 } - 8 x + 16 } { x ^ { 2 } + 2 x } \div ( \frac { 12 } { x + 2 } - x + 2 )$$

Evaluate

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

Short Solution Steps

Express 2\times \frac{x^{2}-8x+16}{x^{2}+2x} as a single fraction.

$$\frac{\frac{2\left(x^{2}-8x+16\right)}{x^{2}+2x}}{\frac{12}{x+2}-x+2}$$

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

$$\frac{\frac{2\left(x^{2}-8x+16\right)}{x^{2}+2x}}{\frac{12}{x+2}+\frac{\left(-x+2\right)\left(x+2\right)}{x+2}}$$

Since \frac{12}{x+2} and \frac{\left(-x+2\right)\left(x+2\right)}{x+2} have the same denominator, add them by adding their numerators.

$$\frac{\frac{2\left(x^{2}-8x+16\right)}{x^{2}+2x}}{\frac{12+\left(-x+2\right)\left(x+2\right)}{x+2}}$$

Do the multiplications in 12+\left(-x+2\right)\left(x+2\right).

$$\frac{\frac{2\left(x^{2}-8x+16\right)}{x^{2}+2x}}{\frac{12-x^{2}-2x+2x+4}{x+2}}$$

Combine like terms in 12-x^{2}-2x+2x+4.

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

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

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

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

Expand

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

Short Solution Steps

Express 2\times \frac{x^{2}-8x+16}{x^{2}+2x} as a single fraction.

$$\frac{\frac{2\left(x^{2}-8x+16\right)}{x^{2}+2x}}{\frac{12}{x+2}-x+2}$$

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

$$\frac{\frac{2\left(x^{2}-8x+16\right)}{x^{2}+2x}}{\frac{12}{x+2}+\frac{\left(-x+2\right)\left(x+2\right)}{x+2}}$$

Since \frac{12}{x+2} and \frac{\left(-x+2\right)\left(x+2\right)}{x+2} have the same denominator, add them by adding their numerators.

$$\frac{\frac{2\left(x^{2}-8x+16\right)}{x^{2}+2x}}{\frac{12+\left(-x+2\right)\left(x+2\right)}{x+2}}$$

Do the multiplications in 12+\left(-x+2\right)\left(x+2\right).

$$\frac{\frac{2\left(x^{2}-8x+16\right)}{x^{2}+2x}}{\frac{12-x^{2}-2x+2x+4}{x+2}}$$

Combine like terms in 12-x^{2}-2x+2x+4.

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

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

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

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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