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

$$\frac { a ^ { 4 } + a ^ { 3 } - a - 1 } { 1 - a ^ { 3 } b ^ { 2 } - a ^ { 3 } + b ^ { 2 } } \cdot \frac { [ 2 + \frac { b } { a } ( a ^ { 2 } - 1 ) ] ^ { 2 } + [ 2 b - \frac { 1 } { a } ( a ^ { 2 } - 1 ) ] ^ { 2 } } { a ^ { 2 } - a ^ { - 2 } }$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{4}+a^{3}-a-1}{1-a^{3}b^{2}-a^{3}+b^{2}}.

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

Cancel out \left(a-1\right)\left(a^{2}+a+1\right) in both numerator and denominator.

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

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

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

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

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

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

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

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

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

To raise \frac{2a+ba^{2}-b}{a} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

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

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

Since \frac{2ba}{a} and \frac{a^{2}-1}{a} have the same denominator, subtract them by subtracting their numerators.

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

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

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

To raise \frac{2ba-a^{2}+1}{a} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

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

$$\frac{a+1}{-b^{2}-1}\times \frac{\frac{4a^{2}+2a^{3}b-2ab+2a^{3}b+b^{2}a^{4}-b^{2}a^{2}-2ab-b^{2}a^{2}+b^{2}+4b^{2}a^{2}-2ba^{3}+2ba-2ba^{3}+a^{4}-a^{2}+2ba-a^{2}+1}{a^{2}}}{a^{2}-a^{-2}}$$

Combine like terms in 4a^{2}+2a^{3}b-2ab+2a^{3}b+b^{2}a^{4}-b^{2}a^{2}-2ab-b^{2}a^{2}+b^{2}+4b^{2}a^{2}-2ba^{3}+2ba-2ba^{3}+a^{4}-a^{2}+2ba-a^{2}+1.

$$\frac{a+1}{-b^{2}-1}\times \frac{\frac{2a^{2}+b^{2}a^{4}+b^{2}+2b^{2}a^{2}+a^{4}+1}{a^{2}}}{a^{2}-a^{-2}}$$

Express \frac{\frac{2a^{2}+b^{2}a^{4}+b^{2}+2b^{2}a^{2}+a^{4}+1}{a^{2}}}{a^{2}-a^{-2}} as a single fraction.

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

Factor the expressions that are not already factored in \frac{2a^{2}+b^{2}a^{4}+b^{2}+2b^{2}a^{2}+a^{4}+1}{a^{2}\left(a^{2}-a^{-2}\right)}.

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

Cancel out a^{2}+1 in both numerator and denominator.

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

Multiply a^{-2} and a^{2} to get 1.

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

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

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

Extract the negative sign in b^{2}+1.

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

Cancel out \left(a+1\right)\left(-b^{2}-1\right) in both numerator and denominator.

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

To find the opposite of a^{2}+1, find the opposite of each term.

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{4}+a^{3}-a-1}{1-a^{3}b^{2}-a^{3}+b^{2}}.

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

Cancel out \left(a-1\right)\left(a^{2}+a+1\right) in both numerator and denominator.

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

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

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

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

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

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

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

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

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

To raise \frac{2a+ba^{2}-b}{a} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

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

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

Since \frac{2ba}{a} and \frac{a^{2}-1}{a} have the same denominator, subtract them by subtracting their numerators.

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

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

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

To raise \frac{2ba-a^{2}+1}{a} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

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

$$\frac{a+1}{-b^{2}-1}\times \frac{\frac{4a^{2}+2a^{3}b-2ab+2a^{3}b+b^{2}a^{4}-b^{2}a^{2}-2ab-b^{2}a^{2}+b^{2}+4b^{2}a^{2}-2ba^{3}+2ba-2ba^{3}+a^{4}-a^{2}+2ba-a^{2}+1}{a^{2}}}{a^{2}-a^{-2}}$$

Combine like terms in 4a^{2}+2a^{3}b-2ab+2a^{3}b+b^{2}a^{4}-b^{2}a^{2}-2ab-b^{2}a^{2}+b^{2}+4b^{2}a^{2}-2ba^{3}+2ba-2ba^{3}+a^{4}-a^{2}+2ba-a^{2}+1.

$$\frac{a+1}{-b^{2}-1}\times \frac{\frac{2a^{2}+b^{2}a^{4}+b^{2}+2b^{2}a^{2}+a^{4}+1}{a^{2}}}{a^{2}-a^{-2}}$$

Express \frac{\frac{2a^{2}+b^{2}a^{4}+b^{2}+2b^{2}a^{2}+a^{4}+1}{a^{2}}}{a^{2}-a^{-2}} as a single fraction.

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

Factor the expressions that are not already factored in \frac{2a^{2}+b^{2}a^{4}+b^{2}+2b^{2}a^{2}+a^{4}+1}{a^{2}\left(a^{2}-a^{-2}\right)}.

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

Cancel out a^{2}+1 in both numerator and denominator.

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

Multiply a^{-2} and a^{2} to get 1.

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

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

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

Extract the negative sign in b^{2}+1.

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

Cancel out \left(a+1\right)\left(-b^{2}-1\right) in both numerator and denominator.

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

To find the opposite of a^{2}+1, find the opposite of each term.

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