Solve (a/a-2b+8b^2/a^2-4b^2+2b/a+2b):a+2b/4b-2a

$$( \frac { a } { a - 2 b } + \frac { 8 b ^ { 2 } } { a ^ { 2 } - 4 b ^ { 2 } } + \frac { 2 b } { a + 2 b } ) : \frac { a + 2 b } { 4 b - 2 a }$$

Evaluate

$$-2$$

Short Solution Steps

Factor a^{2}-4b^{2}.

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

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

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

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

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

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

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

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 a+2b is \left(a-2b\right)\left(a+2b\right). Multiply \frac{2b}{a+2b} times \frac{a-2b}{a-2b}.

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

Since \frac{a^{2}+2ab+8b^{2}}{\left(a-2b\right)\left(a+2b\right)} and \frac{2b\left(a-2b\right)}{\left(a-2b\right)\left(a+2b\right)} have the same denominator, add them by adding their numerators.

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

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

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

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

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

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

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

Cancel out a+2b in both numerator and denominator.

$$\frac{\frac{a+2b}{a-2b}}{\frac{a+2b}{4b-2a}}$$

Divide \frac{a+2b}{a-2b} by \frac{a+2b}{4b-2a} by multiplying \frac{a+2b}{a-2b} by the reciprocal of \frac{a+2b}{4b-2a}.

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

Cancel out a+2b in both numerator and denominator.

$$\frac{-2a+4b}{a-2b}$$

Factor the expressions that are not already factored.

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

Extract the negative sign in 2b-a.

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

Cancel out a-2b in both numerator and denominator.

$$-2$$

Factor

$$-2$$