Solve left(3/a+1-aleft(a+1right)/a+1+a+1/a+1right)a+1/left(a-2right)^2+frac{4}{a-2}-a

$$\left( \frac{ 3 }{ a+1 } - \frac{ a \left( a+1 \right) }{ a+1 } + \frac{ a+1 }{ a+1 } \right) \frac{ a+1 }{ { \left(a-2 \right) }^{ 2 } } + \frac{ 4 }{ a-2 } -a$$

Evaluate

$$-a-1$$

Short Solution Steps

Divide a+1 by a+1 to get 1.

$$\left(\frac{3}{a+1}-\frac{a\left(a+1\right)}{a+1}+1\right)\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{4}{a-2}-a$$

Cancel out a+1 in both numerator and denominator.

$$\left(\frac{3}{a+1}-a+1\right)\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{4}{a-2}-a$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -a+1 times \frac{a+1}{a+1}.

$$\left(\frac{3}{a+1}+\frac{\left(-a+1\right)\left(a+1\right)}{a+1}\right)\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{4}{a-2}-a$$

Since \frac{3}{a+1} and \frac{\left(-a+1\right)\left(a+1\right)}{a+1} have the same denominator, add them by adding their numerators.

$$\frac{3+\left(-a+1\right)\left(a+1\right)}{a+1}\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{4}{a-2}-a$$

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

$$\frac{3-a^{2}-a+a+1}{a+1}\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{4}{a-2}-a$$

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

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

Multiply \frac{4-a^{2}}{a+1} times \frac{a+1}{\left(a-2\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out a+1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-2\right)^{2} and a-2 is \left(a-2\right)^{2}. Multiply \frac{4}{a-2} times \frac{a-2}{a-2}.

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

Since \frac{-a^{2}+4}{\left(a-2\right)^{2}} and \frac{4\left(a-2\right)}{\left(a-2\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -a^{2}+4+4\left(a-2\right).

$$\frac{-a^{2}+4+4a-8}{\left(a-2\right)^{2}}-a$$

Combine like terms in -a^{2}+4+4a-8.

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

Factor the expressions that are not already factored in \frac{-a^{2}-4+4a}{\left(a-2\right)^{2}}.

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

Cancel out a-2 in both numerator and denominator.

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

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

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

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

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

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

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

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

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

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

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

Cancel out a-2 in both numerator and denominator.

$$-a-1$$

Expand

$$-a-1$$

Short Solution Steps

Divide a+1 by a+1 to get 1.

$$\left(\frac{3}{a+1}-\frac{a\left(a+1\right)}{a+1}+1\right)\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{4}{a-2}-a$$

Cancel out a+1 in both numerator and denominator.

$$\left(\frac{3}{a+1}-a+1\right)\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{4}{a-2}-a$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -a+1 times \frac{a+1}{a+1}.

$$\left(\frac{3}{a+1}+\frac{\left(-a+1\right)\left(a+1\right)}{a+1}\right)\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{4}{a-2}-a$$

Since \frac{3}{a+1} and \frac{\left(-a+1\right)\left(a+1\right)}{a+1} have the same denominator, add them by adding their numerators.

$$\frac{3+\left(-a+1\right)\left(a+1\right)}{a+1}\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{4}{a-2}-a$$

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

$$\frac{3-a^{2}-a+a+1}{a+1}\times \frac{a+1}{\left(a-2\right)^{2}}+\frac{4}{a-2}-a$$

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

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

Multiply \frac{4-a^{2}}{a+1} times \frac{a+1}{\left(a-2\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out a+1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-2\right)^{2} and a-2 is \left(a-2\right)^{2}. Multiply \frac{4}{a-2} times \frac{a-2}{a-2}.

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

Since \frac{-a^{2}+4}{\left(a-2\right)^{2}} and \frac{4\left(a-2\right)}{\left(a-2\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -a^{2}+4+4\left(a-2\right).

$$\frac{-a^{2}+4+4a-8}{\left(a-2\right)^{2}}-a$$

Combine like terms in -a^{2}+4+4a-8.

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

Factor the expressions that are not already factored in \frac{-a^{2}-4+4a}{\left(a-2\right)^{2}}.

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

Cancel out a-2 in both numerator and denominator.

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

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

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

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

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

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

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

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

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

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

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

Cancel out a-2 in both numerator and denominator.

$$-a-1$$