Solve a^2-a/a+1-a^2+a/a-1/frac{1{a+1}-1/a-1+frac{2a}{1-a^2}}+frac{2a+1-a^2}{a+1}

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

Evaluate

$$a+1$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+1 and a-1 is \left(a-1\right)\left(a+1\right). Multiply \frac{a^{2}-a}{a+1} times \frac{a-1}{a-1}. Multiply \frac{a^{2}+a}{a-1} times \frac{a+1}{a+1}.

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

Since \frac{\left(a^{2}-a\right)\left(a-1\right)}{\left(a-1\right)\left(a+1\right)} and \frac{\left(a^{2}+a\right)\left(a+1\right)}{\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(a^{2}-a\right)\left(a-1\right)-\left(a^{2}+a\right)\left(a+1\right).

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+1 and a-1 is \left(a-1\right)\left(a+1\right). Multiply \frac{1}{a+1} times \frac{a-1}{a-1}. Multiply \frac{1}{a-1} times \frac{a+1}{a+1}.

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

Since \frac{a-1}{\left(a-1\right)\left(a+1\right)} and \frac{a+1}{\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

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

Combine like terms in a-1-a-1.

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

Factor 1-a^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-1\right)\left(a+1\right) and \left(a-1\right)\left(-a-1\right) is \left(a-1\right)\left(a+1\right). Multiply \frac{2a}{\left(a-1\right)\left(-a-1\right)} times \frac{-1}{-1}.

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

Since \frac{-2}{\left(a-1\right)\left(a+1\right)} and \frac{-2a}{\left(a-1\right)\left(a+1\right)} have the same denominator, add them by adding their numerators.

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

Factor the expressions that are not already factored in \frac{-2-2a}{\left(a-1\right)\left(a+1\right)}.

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

Extract the negative sign in -1-a.

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

Cancel out a+1 in both numerator and denominator.

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

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

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

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

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

Cancel out -1 in both numerator and denominator.

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

Since \frac{2a^{2}}{a+1} and \frac{2a+1-a^{2}}{a+1} have the same denominator, add them by adding their numerators.

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

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

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

Factor the expressions that are not already factored in \frac{a^{2}+2a+1}{a+1}.

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

Cancel out a+1 in both numerator and denominator.

$$a+1$$

Expand

$$a+1$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+1 and a-1 is \left(a-1\right)\left(a+1\right). Multiply \frac{a^{2}-a}{a+1} times \frac{a-1}{a-1}. Multiply \frac{a^{2}+a}{a-1} times \frac{a+1}{a+1}.

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

Since \frac{\left(a^{2}-a\right)\left(a-1\right)}{\left(a-1\right)\left(a+1\right)} and \frac{\left(a^{2}+a\right)\left(a+1\right)}{\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(a^{2}-a\right)\left(a-1\right)-\left(a^{2}+a\right)\left(a+1\right).

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+1 and a-1 is \left(a-1\right)\left(a+1\right). Multiply \frac{1}{a+1} times \frac{a-1}{a-1}. Multiply \frac{1}{a-1} times \frac{a+1}{a+1}.

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

Since \frac{a-1}{\left(a-1\right)\left(a+1\right)} and \frac{a+1}{\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

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

Combine like terms in a-1-a-1.

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

Factor 1-a^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-1\right)\left(a+1\right) and \left(a-1\right)\left(-a-1\right) is \left(a-1\right)\left(a+1\right). Multiply \frac{2a}{\left(a-1\right)\left(-a-1\right)} times \frac{-1}{-1}.

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

Since \frac{-2}{\left(a-1\right)\left(a+1\right)} and \frac{-2a}{\left(a-1\right)\left(a+1\right)} have the same denominator, add them by adding their numerators.

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

Factor the expressions that are not already factored in \frac{-2-2a}{\left(a-1\right)\left(a+1\right)}.

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

Extract the negative sign in -1-a.

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

Cancel out a+1 in both numerator and denominator.

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

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

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

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

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

Cancel out -1 in both numerator and denominator.

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

Since \frac{2a^{2}}{a+1} and \frac{2a+1-a^{2}}{a+1} have the same denominator, add them by adding their numerators.

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

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

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

Factor the expressions that are not already factored in \frac{a^{2}+2a+1}{a+1}.

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

Cancel out a+1 in both numerator and denominator.

$$a+1$$