Solve (3/x-1-x-1)divx-2/x^2-2x+1

$$( \frac { 3 } { x - 1 } - x - 1 ) \div \frac { x - 2 } { x ^ { 2 } - 2 x + 1 }$$

Evaluate

$$-\left(x-1\right)\left(x+2\right)$$

Short Solution Steps

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

$$\frac{\frac{3}{x-1}+\frac{\left(-x-1\right)\left(x-1\right)}{x-1}}{\frac{x-2}{x^{2}-2x+1}}$$

Since \frac{3}{x-1} and \frac{\left(-x-1\right)\left(x-1\right)}{x-1} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3+\left(-x-1\right)\left(x-1\right)}{x-1}}{\frac{x-2}{x^{2}-2x+1}}$$

Do the multiplications in 3+\left(-x-1\right)\left(x-1\right).

$$\frac{\frac{3-x^{2}+x-x+1}{x-1}}{\frac{x-2}{x^{2}-2x+1}}$$

Combine like terms in 3-x^{2}+x-x+1.

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

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

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

Factor the expressions that are not already factored.

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

Cancel out \left(x-2\right)\left(x-1\right) in both numerator and denominator.

$$\left(x-1\right)\left(-x-2\right)$$

Expand the expression.

$$-x^{2}-x+2$$

Expand

$$2-x-x^{2}$$

Short Solution Steps

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

$$\frac{\frac{3}{x-1}+\frac{\left(-x-1\right)\left(x-1\right)}{x-1}}{\frac{x-2}{x^{2}-2x+1}}$$

Since \frac{3}{x-1} and \frac{\left(-x-1\right)\left(x-1\right)}{x-1} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3+\left(-x-1\right)\left(x-1\right)}{x-1}}{\frac{x-2}{x^{2}-2x+1}}$$

Do the multiplications in 3+\left(-x-1\right)\left(x-1\right).

$$\frac{\frac{3-x^{2}+x-x+1}{x-1}}{\frac{x-2}{x^{2}-2x+1}}$$

Combine like terms in 3-x^{2}+x-x+1.

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

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

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

Factor the expressions that are not already factored.

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

Cancel out \left(x-2\right)\left(x-1\right) in both numerator and denominator.

$$\left(x-1\right)\left(-x-2\right)$$

Expand the expression.

$$-x^{2}-x+2$$