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

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

Evaluate

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

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}.

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

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

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

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

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

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

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

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

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

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

Expand

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

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}.

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

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

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

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

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

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

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

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

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

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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