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

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

Evaluate

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

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}.

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

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.

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

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

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

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

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

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

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

Cancel out x+1 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x-2 in both numerator and denominator.

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

Expand

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

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}.

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

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.

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

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

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

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

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

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

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

Cancel out x+1 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x-2 in both numerator and denominator.

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