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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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