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

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

Evaluate

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

Short Solution Steps

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

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

Since \frac{3}{x+2} and \frac{\left(x-2\right)\left(x+2\right)}{x+2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3+\left(x-2\right)\left(x+2\right).

$$\frac{\frac{3+x^{2}+2x-2x-4}{x+2}}{\frac{x^{2}-2x+1}{x+2}}$$

Combine like terms in 3+x^{2}+2x-2x-4.

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

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

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

Cancel out x+2 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x-1 in both numerator and denominator.

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

Expand

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

Short Solution Steps

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

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

Since \frac{3}{x+2} and \frac{\left(x-2\right)\left(x+2\right)}{x+2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3+\left(x-2\right)\left(x+2\right).

$$\frac{\frac{3+x^{2}+2x-2x-4}{x+2}}{\frac{x^{2}-2x+1}{x+2}}$$

Combine like terms in 3+x^{2}+2x-2x-4.

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

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

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

Cancel out x+2 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x-1 in both numerator and denominator.

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