Solve a+1+a^2/a^2-1+a^2+2a+1/a-1*(a^2-4)

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

Evaluate

$$\frac{a^{5}+3a^{4}-a^{3}-10a^{2}-11a-3}{a^{2}-1}$$

Short Solution Steps

Express \frac{a^{2}+2a+1}{a-1}\left(a^{2}-4\right) as a single fraction.

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

Factor a^{2}-1.

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

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

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

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

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

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

$$\frac{a+1+a^{2}+a^{5}+a^{4}-4a^{3}-4a^{2}+2a^{4}+2a^{3}-8a^{2}-8a+a^{3}+a^{2}-4a-4}{\left(a-1\right)\left(a+1\right)}$$

Combine like terms in a+1+a^{2}+a^{5}+a^{4}-4a^{3}-4a^{2}+2a^{4}+2a^{3}-8a^{2}-8a+a^{3}+a^{2}-4a-4.

$$\frac{-11a-3-10a^{2}+a^{5}+3a^{4}-a^{3}}{\left(a-1\right)\left(a+1\right)}$$

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

$$\frac{-11a-3-10a^{2}+a^{5}+3a^{4}-a^{3}}{a^{2}-1}$$

Expand

$$\frac{a^{5}+3a^{4}-a^{3}-10a^{2}-11a-3}{a^{2}-1}$$

Short Solution Steps

Express \frac{a^{2}+2a+1}{a-1}\left(a^{2}-4\right) as a single fraction.

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

Factor a^{2}-1.

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

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

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

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

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

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

$$\frac{a+1+a^{2}+a^{5}+a^{4}-4a^{3}-4a^{2}+2a^{4}+2a^{3}-8a^{2}-8a+a^{3}+a^{2}-4a-4}{\left(a-1\right)\left(a+1\right)}$$

Combine like terms in a+1+a^{2}+a^{5}+a^{4}-4a^{3}-4a^{2}+2a^{4}+2a^{3}-8a^{2}-8a+a^{3}+a^{2}-4a-4.

$$\frac{-11a-3-10a^{2}+a^{5}+3a^{4}-a^{3}}{\left(a-1\right)\left(a+1\right)}$$

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

$$\frac{-11a-3-10a^{2}+a^{5}+3a^{4}-a^{3}}{a^{2}-1}$$