Solve left(1/2-a+6a-4-a^2/a^3-8-2-a/a^2+2a+4right)frac{a^3+4{a}^2+8a+8}{4-4a+{a}^2-{a}^3}

$$\left( \frac{ 1 }{ 2-a } + \frac{ 6a-4- { a }^{ 2 } }{ { a }^{ 3 } -8 } - \frac{ 2-a }{ { a }^{ 2 } +2a+4 } \right) \frac{ { a }^{ 3 } +4 { a }^{ 2 } +8a+8 }{ 4-4a+ { a }^{ 2 } - { a }^{ 3 } }$$

Evaluate

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

Short Solution Steps

Factor a^{3}-8.

$$\left(\frac{1}{2-a}+\frac{6a-4-a^{2}}{\left(a-2\right)\left(a^{2}+2a+4\right)}-\frac{2-a}{a^{2}+2a+4}\right)\times \frac{a^{3}+4a^{2}+8a+8}{4-4a+a^{2}-a^{3}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2-a and \left(a-2\right)\left(a^{2}+2a+4\right) is \left(a-2\right)\left(a^{2}+2a+4\right). Multiply \frac{1}{2-a} times \frac{-\left(a^{2}+2a+4\right)}{-\left(a^{2}+2a+4\right)}.

$$\left(\frac{-\left(a^{2}+2a+4\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)}+\frac{6a-4-a^{2}}{\left(a-2\right)\left(a^{2}+2a+4\right)}-\frac{2-a}{a^{2}+2a+4}\right)\times \frac{a^{3}+4a^{2}+8a+8}{4-4a+a^{2}-a^{3}}$$

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

$$\left(\frac{-\left(a^{2}+2a+4\right)+6a-4-a^{2}}{\left(a-2\right)\left(a^{2}+2a+4\right)}-\frac{2-a}{a^{2}+2a+4}\right)\times \frac{a^{3}+4a^{2}+8a+8}{4-4a+a^{2}-a^{3}}$$

Do the multiplications in -\left(a^{2}+2a+4\right)+6a-4-a^{2}.

$$\left(\frac{-a^{2}-2a-4+6a-4-a^{2}}{\left(a-2\right)\left(a^{2}+2a+4\right)}-\frac{2-a}{a^{2}+2a+4}\right)\times \frac{a^{3}+4a^{2}+8a+8}{4-4a+a^{2}-a^{3}}$$

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-2\right)\left(a^{2}+2a+4\right) and a^{2}+2a+4 is \left(a-2\right)\left(a^{2}+2a+4\right). Multiply \frac{2-a}{a^{2}+2a+4} times \frac{a-2}{a-2}.

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

Since \frac{-2a^{2}+4a-8}{\left(a-2\right)\left(a^{2}+2a+4\right)} and \frac{\left(2-a\right)\left(a-2\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -2a^{2}+4a-8-\left(2-a\right)\left(a-2\right).

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

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

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

Multiply \frac{-a^{2}-4}{\left(a-2\right)\left(a^{2}+2a+4\right)} times \frac{a^{3}+4a^{2}+8a+8}{4-4a+a^{2}-a^{3}} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in -a^{2}-4.

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

Cancel out \left(a^{2}+4\right)\left(a^{2}+2a+4\right) in both numerator and denominator.

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

Expand the expression.

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

Expand

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

Short Solution Steps

Factor a^{3}-8.

$$\left(\frac{1}{2-a}+\frac{6a-4-a^{2}}{\left(a-2\right)\left(a^{2}+2a+4\right)}-\frac{2-a}{a^{2}+2a+4}\right)\times \frac{a^{3}+4a^{2}+8a+8}{4-4a+a^{2}-a^{3}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2-a and \left(a-2\right)\left(a^{2}+2a+4\right) is \left(a-2\right)\left(a^{2}+2a+4\right). Multiply \frac{1}{2-a} times \frac{-\left(a^{2}+2a+4\right)}{-\left(a^{2}+2a+4\right)}.

$$\left(\frac{-\left(a^{2}+2a+4\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)}+\frac{6a-4-a^{2}}{\left(a-2\right)\left(a^{2}+2a+4\right)}-\frac{2-a}{a^{2}+2a+4}\right)\times \frac{a^{3}+4a^{2}+8a+8}{4-4a+a^{2}-a^{3}}$$

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

$$\left(\frac{-\left(a^{2}+2a+4\right)+6a-4-a^{2}}{\left(a-2\right)\left(a^{2}+2a+4\right)}-\frac{2-a}{a^{2}+2a+4}\right)\times \frac{a^{3}+4a^{2}+8a+8}{4-4a+a^{2}-a^{3}}$$

Do the multiplications in -\left(a^{2}+2a+4\right)+6a-4-a^{2}.

$$\left(\frac{-a^{2}-2a-4+6a-4-a^{2}}{\left(a-2\right)\left(a^{2}+2a+4\right)}-\frac{2-a}{a^{2}+2a+4}\right)\times \frac{a^{3}+4a^{2}+8a+8}{4-4a+a^{2}-a^{3}}$$

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-2\right)\left(a^{2}+2a+4\right) and a^{2}+2a+4 is \left(a-2\right)\left(a^{2}+2a+4\right). Multiply \frac{2-a}{a^{2}+2a+4} times \frac{a-2}{a-2}.

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

Since \frac{-2a^{2}+4a-8}{\left(a-2\right)\left(a^{2}+2a+4\right)} and \frac{\left(2-a\right)\left(a-2\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -2a^{2}+4a-8-\left(2-a\right)\left(a-2\right).

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

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

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

Multiply \frac{-a^{2}-4}{\left(a-2\right)\left(a^{2}+2a+4\right)} times \frac{a^{3}+4a^{2}+8a+8}{4-4a+a^{2}-a^{3}} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in -a^{2}-4.

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

Cancel out \left(a^{2}+4\right)\left(a^{2}+2a+4\right) in both numerator and denominator.

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

Expand the expression.

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