Solve (-3/a+1-a(a+1)/a+1+a+1/a+1)*a+1/(a-2)^2+frac{4}{a-2}-a

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

Evaluate

$$\frac{-a^{3}+3a^{2}-10}{\left(a-2\right)^{2}}$$

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.

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

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

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

Cancel out a+1 in both numerator and denominator.

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

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}.

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

Since \frac{-a^{2}-2}{\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.

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

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

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

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

$$\frac{-a^{2}-10+4a}{\left(a-2\right)^{2}}-a$$

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

$$\frac{-a^{2}-10+4a}{\left(a-2\right)^{2}}-\frac{a\left(a-2\right)^{2}}{\left(a-2\right)^{2}}$$

Since \frac{-a^{2}-10+4a}{\left(a-2\right)^{2}} and \frac{a\left(a-2\right)^{2}}{\left(a-2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-a^{2}-10+4a-a\left(a-2\right)^{2}}{\left(a-2\right)^{2}}$$

Do the multiplications in -a^{2}-10+4a-a\left(a-2\right)^{2}.

$$\frac{-a^{2}-10+4a-a^{3}+4a^{2}-4a}{\left(a-2\right)^{2}}$$

Combine like terms in -a^{2}-10+4a-a^{3}+4a^{2}-4a.

$$\frac{3a^{2}-10-a^{3}}{\left(a-2\right)^{2}}$$

Expand \left(a-2\right)^{2}.

$$\frac{3a^{2}-10-a^{3}}{a^{2}-4a+4}$$

Expand

$$\frac{-a^{3}+3a^{2}-10}{\left(a-2\right)^{2}}$$

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.

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

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

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

Cancel out a+1 in both numerator and denominator.

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

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}.

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

Since \frac{-a^{2}-2}{\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.

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

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

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

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

$$\frac{-a^{2}-10+4a}{\left(a-2\right)^{2}}-a$$

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

$$\frac{-a^{2}-10+4a}{\left(a-2\right)^{2}}-\frac{a\left(a-2\right)^{2}}{\left(a-2\right)^{2}}$$

Since \frac{-a^{2}-10+4a}{\left(a-2\right)^{2}} and \frac{a\left(a-2\right)^{2}}{\left(a-2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-a^{2}-10+4a-a\left(a-2\right)^{2}}{\left(a-2\right)^{2}}$$

Do the multiplications in -a^{2}-10+4a-a\left(a-2\right)^{2}.

$$\frac{-a^{2}-10+4a-a^{3}+4a^{2}-4a}{\left(a-2\right)^{2}}$$

Combine like terms in -a^{2}-10+4a-a^{3}+4a^{2}-4a.

$$\frac{3a^{2}-10-a^{3}}{\left(a-2\right)^{2}}$$

Expand \left(a-2\right)^{2}.

$$\frac{3a^{2}-10-a^{3}}{a^{2}-4a+4}$$