Solve -a-b/a+2bdiva^2-b^2/a^2+4ab+4b^2+2

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

Evaluate

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

Short Solution Steps

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

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

Express \left(-\frac{a-b}{a+2b}\right)\left(a^{2}+4ab+4b^{2}+2\right) as a single fraction.

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

Express \frac{\frac{-\left(a-b\right)\left(a^{2}+4ab+4b^{2}+2\right)}{a+2b}}{a^{2}-b^{2}} as a single fraction.

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

Factor the expressions that are not already factored.

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

Cancel out a-b in both numerator and denominator.

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

Expand the expression.

$$\frac{-a^{2}-4ab-4b^{2}-2}{a^{2}+3ab+2b^{2}}$$

Expand

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

Short Solution Steps

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

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

Express \left(-\frac{a-b}{a+2b}\right)\left(a^{2}+4ab+4b^{2}+2\right) as a single fraction.

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

Express \frac{\frac{-\left(a-b\right)\left(a^{2}+4ab+4b^{2}+2\right)}{a+2b}}{a^{2}-b^{2}} as a single fraction.

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

Factor the expressions that are not already factored.

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

Cancel out a-b in both numerator and denominator.

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

Expand the expression.

$$\frac{-a^{2}-4ab-4b^{2}-2}{a^{2}+3ab+2b^{2}}$$