Solve (a/a-2b-2a+2b/a+2b+a/b+1)*a^2b-4b^3/a^3+a^2b

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

Evaluate

$$\frac{a}{a+b}$$

Short Solution Steps

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

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

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

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

Do the multiplications in a\left(a+2b\right)-\left(2a+2b\right)\left(a-2b\right).

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

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

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

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

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{b\left(a-2b\right)\left(a+2b\right)}{b\left(a-2b\right)\left(a+2b\right)}.

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

Since \frac{-a^{2}b+4b^{3}+a^{3}}{b\left(a-2b\right)\left(a+2b\right)} and \frac{b\left(a-2b\right)\left(a+2b\right)}{b\left(a-2b\right)\left(a+2b\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -a^{2}b+4b^{3}+a^{3}+b\left(a-2b\right)\left(a+2b\right).

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

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

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

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

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

Factor the expressions that are not already factored.

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

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

$$\frac{a}{a+b}$$

Expand

$$\frac{a}{a+b}$$

Short Solution Steps

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

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

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

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

Do the multiplications in a\left(a+2b\right)-\left(2a+2b\right)\left(a-2b\right).

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

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

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

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

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{b\left(a-2b\right)\left(a+2b\right)}{b\left(a-2b\right)\left(a+2b\right)}.

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

Since \frac{-a^{2}b+4b^{3}+a^{3}}{b\left(a-2b\right)\left(a+2b\right)} and \frac{b\left(a-2b\right)\left(a+2b\right)}{b\left(a-2b\right)\left(a+2b\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -a^{2}b+4b^{3}+a^{3}+b\left(a-2b\right)\left(a+2b\right).

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

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

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

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

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

Factor the expressions that are not already factored.

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

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

$$\frac{a}{a+b}$$