Solve frac{a^{2}-b^2}{a+2b}:b-a/a^2+2ab+a^3+b^3/a+b

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

Evaluate

$$b\left(b-2a\right)$$

Short Solution Steps

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

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

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

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

Extract the negative sign in a-b.

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

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

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

Expand the expression.

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

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

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

Cancel out a+b in both numerator and denominator.

$$-a^{2}-ab+a^{2}-ab+b^{2}$$

Combine -a^{2} and a^{2} to get 0.

$$-ab-ab+b^{2}$$

Combine -ab and -ab to get -2ab.

$$-2ab+b^{2}$$

Expand

$$b^{2}-2ab$$

Short Solution Steps

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

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

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

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

Extract the negative sign in a-b.

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

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

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

Expand the expression.

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

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

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

Cancel out a+b in both numerator and denominator.

$$-a^{2}-ab+a^{2}-ab+b^{2}$$

Combine -a^{2} and a^{2} to get 0.

$$-ab-ab+b^{2}$$

Combine -ab and -ab to get -2ab.

$$-2ab+b^{2}$$