Solve (x^2-2x+4/x-1-x+2)divx^2+4x+4/1-x

$$( \frac { x ^ { 2 } - 2 x + 4 } { x - 1 } - x + 2 ) \div \frac { x ^ { 2 } + 4 x + 4 } { 1 - x }$$

Evaluate

$$-\frac{1}{x+2}$$

Short Solution Steps

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

$$\frac{\frac{x^{2}-2x+4}{x-1}+\frac{\left(-x+2\right)\left(x-1\right)}{x-1}}{\frac{x^{2}+4x+4}{1-x}}$$

Since \frac{x^{2}-2x+4}{x-1} and \frac{\left(-x+2\right)\left(x-1\right)}{x-1} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x^{2}-2x+4+\left(-x+2\right)\left(x-1\right)}{x-1}}{\frac{x^{2}+4x+4}{1-x}}$$

Do the multiplications in x^{2}-2x+4+\left(-x+2\right)\left(x-1\right).

$$\frac{\frac{x^{2}-2x+4-x^{2}+x+2x-2}{x-1}}{\frac{x^{2}+4x+4}{1-x}}$$

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

$$\frac{\frac{x+2}{x-1}}{\frac{x^{2}+4x+4}{1-x}}$$

Divide \frac{x+2}{x-1} by \frac{x^{2}+4x+4}{1-x} by multiplying \frac{x+2}{x-1} by the reciprocal of \frac{x^{2}+4x+4}{1-x}.

$$\frac{\left(x+2\right)\left(1-x\right)}{\left(x-1\right)\left(x^{2}+4x+4\right)}$$

Extract the negative sign in 1-x.

$$\frac{-\left(x-1\right)\left(x+2\right)}{\left(x-1\right)\left(x^{2}+4x+4\right)}$$

Cancel out x-1 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x+2 in both numerator and denominator.

$$\frac{-1}{x+2}$$

Expand

$$-\frac{1}{x+2}$$

Short Solution Steps

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

$$\frac{\frac{x^{2}-2x+4}{x-1}+\frac{\left(-x+2\right)\left(x-1\right)}{x-1}}{\frac{x^{2}+4x+4}{1-x}}$$

Since \frac{x^{2}-2x+4}{x-1} and \frac{\left(-x+2\right)\left(x-1\right)}{x-1} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x^{2}-2x+4+\left(-x+2\right)\left(x-1\right)}{x-1}}{\frac{x^{2}+4x+4}{1-x}}$$

Do the multiplications in x^{2}-2x+4+\left(-x+2\right)\left(x-1\right).

$$\frac{\frac{x^{2}-2x+4-x^{2}+x+2x-2}{x-1}}{\frac{x^{2}+4x+4}{1-x}}$$

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

$$\frac{\frac{x+2}{x-1}}{\frac{x^{2}+4x+4}{1-x}}$$

Divide \frac{x+2}{x-1} by \frac{x^{2}+4x+4}{1-x} by multiplying \frac{x+2}{x-1} by the reciprocal of \frac{x^{2}+4x+4}{1-x}.

$$\frac{\left(x+2\right)\left(1-x\right)}{\left(x-1\right)\left(x^{2}+4x+4\right)}$$

Extract the negative sign in 1-x.

$$\frac{-\left(x-1\right)\left(x+2\right)}{\left(x-1\right)\left(x^{2}+4x+4\right)}$$

Cancel out x-1 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x+2 in both numerator and denominator.

$$\frac{-1}{x+2}$$