Solve (2a^2-2/a^2+ab:1-a/a+b)*1/a^3+1=

$$( \frac { 2 a ^ { 2 } - 2 } { a ^ { 2 } + a b } : \frac { 1 - a } { a + b } ) \cdot \frac { 1 } { a ^ { 3 } + 1 } =$$

Evaluate

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

Short Solution Steps

Divide \frac{2a^{2}-2}{a^{2}+ab} by \frac{1-a}{a+b} by multiplying \frac{2a^{2}-2}{a^{2}+ab} by the reciprocal of \frac{1-a}{a+b}.

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

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

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

Extract the negative sign in -1+a.

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

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

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

Multiply \frac{-2\left(a+1\right)}{a} times \frac{1}{a^{3}+1} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored.

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

Cancel out a+1 in both numerator and denominator.

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

Expand the expression.

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

Expand

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

Short Solution Steps

Divide \frac{2a^{2}-2}{a^{2}+ab} by \frac{1-a}{a+b} by multiplying \frac{2a^{2}-2}{a^{2}+ab} by the reciprocal of \frac{1-a}{a+b}.

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

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

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

Extract the negative sign in -1+a.

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

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

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

Multiply \frac{-2\left(a+1\right)}{a} times \frac{1}{a^{3}+1} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored.

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

Cancel out a+1 in both numerator and denominator.

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

Expand the expression.

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