Solve {l}{a-1/a^2+2a+1}{-3/2a-2+a/a^2-1}

$$\left. \begin{array} { l } { \frac { a - 1 } { a ^ { 2 } + 2 a + 1 } } \\ { - \frac { 3 } { 2 a - 2 } + \frac { a } { a ^ { 2 } - 1 } } \end{array} \right.$$

Evaluate

$$\frac{a^{2}-8a-1}{2\left(a-1\right)\left(a+1\right)^{2}}$$

Short Solution Steps

Factor a^{2}+2a+1. Factor 2a-2.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a+1\right)^{2} and 2\left(a-1\right) is 2\left(a-1\right)\left(a+1\right)^{2}. Multiply \frac{a-1}{\left(a+1\right)^{2}} times \frac{2\left(a-1\right)}{2\left(a-1\right)}. Multiply \frac{3}{2\left(a-1\right)} times \frac{\left(a+1\right)^{2}}{\left(a+1\right)^{2}}.

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

Since \frac{\left(a-1\right)\times 2\left(a-1\right)}{2\left(a-1\right)\left(a+1\right)^{2}} and \frac{3\left(a+1\right)^{2}}{2\left(a-1\right)\left(a+1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(a-1\right)\times 2\left(a-1\right)-3\left(a+1\right)^{2}.

$$\frac{2a^{2}-2a-2a+2-3a^{2}-6a-3}{2\left(a-1\right)\left(a+1\right)^{2}}+\frac{a}{a^{2}-1}$$

Combine like terms in 2a^{2}-2a-2a+2-3a^{2}-6a-3.

$$\frac{-a^{2}-10a-1}{2\left(a-1\right)\left(a+1\right)^{2}}+\frac{a}{a^{2}-1}$$

Factor a^{2}-1.

$$\frac{-a^{2}-10a-1}{2\left(a-1\right)\left(a+1\right)^{2}}+\frac{a}{\left(a-1\right)\left(a+1\right)}$$

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

$$\frac{-a^{2}-10a-1}{2\left(a-1\right)\left(a+1\right)^{2}}+\frac{a\times 2\left(a+1\right)}{2\left(a-1\right)\left(a+1\right)^{2}}$$

Since \frac{-a^{2}-10a-1}{2\left(a-1\right)\left(a+1\right)^{2}} and \frac{a\times 2\left(a+1\right)}{2\left(a-1\right)\left(a+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-a^{2}-10a-1+a\times 2\left(a+1\right)}{2\left(a-1\right)\left(a+1\right)^{2}}$$

Do the multiplications in -a^{2}-10a-1+a\times 2\left(a+1\right).

$$\frac{-a^{2}-10a-1+2a^{2}+2a}{2\left(a-1\right)\left(a+1\right)^{2}}$$

Combine like terms in -a^{2}-10a-1+2a^{2}+2a.

$$\frac{a^{2}-8a-1}{2\left(a-1\right)\left(a+1\right)^{2}}$$

Expand 2\left(a-1\right)\left(a+1\right)^{2}.

$$\frac{a^{2}-8a-1}{2a^{3}+2a^{2}-2a-2}$$

Expand

$$\frac{a^{2}-8a-1}{2\left(a-1\right)\left(a+1\right)^{2}}$$

Short Solution Steps

Factor a^{2}+2a+1. Factor 2a-2.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a+1\right)^{2} and 2\left(a-1\right) is 2\left(a-1\right)\left(a+1\right)^{2}. Multiply \frac{a-1}{\left(a+1\right)^{2}} times \frac{2\left(a-1\right)}{2\left(a-1\right)}. Multiply \frac{3}{2\left(a-1\right)} times \frac{\left(a+1\right)^{2}}{\left(a+1\right)^{2}}.

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

Since \frac{\left(a-1\right)\times 2\left(a-1\right)}{2\left(a-1\right)\left(a+1\right)^{2}} and \frac{3\left(a+1\right)^{2}}{2\left(a-1\right)\left(a+1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(a-1\right)\times 2\left(a-1\right)-3\left(a+1\right)^{2}.

$$\frac{2a^{2}-2a-2a+2-3a^{2}-6a-3}{2\left(a-1\right)\left(a+1\right)^{2}}+\frac{a}{a^{2}-1}$$

Combine like terms in 2a^{2}-2a-2a+2-3a^{2}-6a-3.

$$\frac{-a^{2}-10a-1}{2\left(a-1\right)\left(a+1\right)^{2}}+\frac{a}{a^{2}-1}$$

Factor a^{2}-1.

$$\frac{-a^{2}-10a-1}{2\left(a-1\right)\left(a+1\right)^{2}}+\frac{a}{\left(a-1\right)\left(a+1\right)}$$

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

$$\frac{-a^{2}-10a-1}{2\left(a-1\right)\left(a+1\right)^{2}}+\frac{a\times 2\left(a+1\right)}{2\left(a-1\right)\left(a+1\right)^{2}}$$

Since \frac{-a^{2}-10a-1}{2\left(a-1\right)\left(a+1\right)^{2}} and \frac{a\times 2\left(a+1\right)}{2\left(a-1\right)\left(a+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-a^{2}-10a-1+a\times 2\left(a+1\right)}{2\left(a-1\right)\left(a+1\right)^{2}}$$

Do the multiplications in -a^{2}-10a-1+a\times 2\left(a+1\right).

$$\frac{-a^{2}-10a-1+2a^{2}+2a}{2\left(a-1\right)\left(a+1\right)^{2}}$$

Combine like terms in -a^{2}-10a-1+2a^{2}+2a.

$$\frac{a^{2}-8a-1}{2\left(a-1\right)\left(a+1\right)^{2}}$$

Expand 2\left(a-1\right)\left(a+1\right)^{2}.

$$\frac{a^{2}-8a-1}{2a^{3}+2a^{2}-2a-2}$$