Solve {c}{4a^2-4a-1/a^2-1}{+(2+1/a-1)}

$$\left. \begin{array} { c } { \frac { 4 a ^ { 2 } - 4 a - 1 } { a ^ { 2 } - 1 } } \\ { + ( 2 + \frac { 1 } { a - 1 } ) } \end{array} \right.$$

Evaluate

$$\frac{6a^{2}-3a-2}{a^{2}-1}$$

Short Solution Steps

Factor a^{2}-1.

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

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

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

Since \frac{4a^{2}-4a-1}{\left(a-1\right)\left(a+1\right)} and \frac{2\left(a-1\right)\left(a+1\right)}{\left(a-1\right)\left(a+1\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 4a^{2}-4a-1+2\left(a-1\right)\left(a+1\right).

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

Combine like terms in 4a^{2}-4a-1+2a^{2}+2a-2a-2.

$$\frac{6a^{2}-4a-3}{\left(a-1\right)\left(a+1\right)}+\frac{1}{a-1}$$

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 a-1 is \left(a-1\right)\left(a+1\right). Multiply \frac{1}{a-1} times \frac{a+1}{a+1}.

$$\frac{6a^{2}-4a-3}{\left(a-1\right)\left(a+1\right)}+\frac{a+1}{\left(a-1\right)\left(a+1\right)}$$

Since \frac{6a^{2}-4a-3}{\left(a-1\right)\left(a+1\right)} and \frac{a+1}{\left(a-1\right)\left(a+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{6a^{2}-4a-3+a+1}{\left(a-1\right)\left(a+1\right)}$$

Combine like terms in 6a^{2}-4a-3+a+1.

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

Expand \left(a-1\right)\left(a+1\right).

$$\frac{6a^{2}-3a-2}{a^{2}-1}$$

Expand

$$\frac{6a^{2}-3a-2}{a^{2}-1}$$

Short Solution Steps

Factor a^{2}-1.

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

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

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

Since \frac{4a^{2}-4a-1}{\left(a-1\right)\left(a+1\right)} and \frac{2\left(a-1\right)\left(a+1\right)}{\left(a-1\right)\left(a+1\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 4a^{2}-4a-1+2\left(a-1\right)\left(a+1\right).

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

Combine like terms in 4a^{2}-4a-1+2a^{2}+2a-2a-2.

$$\frac{6a^{2}-4a-3}{\left(a-1\right)\left(a+1\right)}+\frac{1}{a-1}$$

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 a-1 is \left(a-1\right)\left(a+1\right). Multiply \frac{1}{a-1} times \frac{a+1}{a+1}.

$$\frac{6a^{2}-4a-3}{\left(a-1\right)\left(a+1\right)}+\frac{a+1}{\left(a-1\right)\left(a+1\right)}$$

Since \frac{6a^{2}-4a-3}{\left(a-1\right)\left(a+1\right)} and \frac{a+1}{\left(a-1\right)\left(a+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{6a^{2}-4a-3+a+1}{\left(a-1\right)\left(a+1\right)}$$

Combine like terms in 6a^{2}-4a-3+a+1.

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

Expand \left(a-1\right)\left(a+1\right).

$$\frac{6a^{2}-3a-2}{a^{2}-1}$$