Solve (a^2+6ab-9b/a+3b+a^2-4ab+4b^2/a-2b)div(4a+2b)

$$( \frac { a ^ { 2 } + 6 a b - 9 b } { a + 3 b } + \frac { a ^ { 2 } - 4 a b + 4 b ^ { 2 } } { a - 2 b } ) \div ( 4 a + 2 b )$$

Evaluate

$$\frac{2a^{2}+7ab-6b^{2}-9b}{2\left(a+3b\right)\left(2a+b\right)}$$

Short Solution Steps

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

$$\frac{\frac{a^{2}+6ab-9b}{a+3b}+\frac{\left(a-2b\right)^{2}}{a-2b}}{4a+2b}$$

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

$$\frac{\frac{a^{2}+6ab-9b}{a+3b}+a-2b}{4a+2b}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a-2b times \frac{a+3b}{a+3b}.

$$\frac{\frac{a^{2}+6ab-9b}{a+3b}+\frac{\left(a-2b\right)\left(a+3b\right)}{a+3b}}{4a+2b}$$

Since \frac{a^{2}+6ab-9b}{a+3b} and \frac{\left(a-2b\right)\left(a+3b\right)}{a+3b} have the same denominator, add them by adding their numerators.

$$\frac{\frac{a^{2}+6ab-9b+\left(a-2b\right)\left(a+3b\right)}{a+3b}}{4a+2b}$$

Do the multiplications in a^{2}+6ab-9b+\left(a-2b\right)\left(a+3b\right).

$$\frac{\frac{a^{2}+6ab-9b+a^{2}+3ab-2ba-6b^{2}}{a+3b}}{4a+2b}$$

Combine like terms in a^{2}+6ab-9b+a^{2}+3ab-2ba-6b^{2}.

$$\frac{\frac{2a^{2}-6b^{2}+7ab-9b}{a+3b}}{4a+2b}$$

Express \frac{\frac{2a^{2}-6b^{2}+7ab-9b}{a+3b}}{4a+2b} as a single fraction.

$$\frac{2a^{2}-6b^{2}+7ab-9b}{\left(a+3b\right)\left(4a+2b\right)}$$

Use the distributive property to multiply a+3b by 4a+2b and combine like terms.

$$\frac{2a^{2}-6b^{2}+7ab-9b}{4a^{2}+14ab+6b^{2}}$$

Expand

$$-\frac{9b+6b^{2}-7ab-2a^{2}}{2\left(a+3b\right)\left(2a+b\right)}$$

Short Solution Steps

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

$$\frac{\frac{a^{2}+6ab-9b}{a+3b}+\frac{\left(a-2b\right)^{2}}{a-2b}}{4a+2b}$$

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

$$\frac{\frac{a^{2}+6ab-9b}{a+3b}+a-2b}{4a+2b}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a-2b times \frac{a+3b}{a+3b}.

$$\frac{\frac{a^{2}+6ab-9b}{a+3b}+\frac{\left(a-2b\right)\left(a+3b\right)}{a+3b}}{4a+2b}$$

Since \frac{a^{2}+6ab-9b}{a+3b} and \frac{\left(a-2b\right)\left(a+3b\right)}{a+3b} have the same denominator, add them by adding their numerators.

$$\frac{\frac{a^{2}+6ab-9b+\left(a-2b\right)\left(a+3b\right)}{a+3b}}{4a+2b}$$

Do the multiplications in a^{2}+6ab-9b+\left(a-2b\right)\left(a+3b\right).

$$\frac{\frac{a^{2}+6ab-9b+a^{2}+3ab-2ba-6b^{2}}{a+3b}}{4a+2b}$$

Combine like terms in a^{2}+6ab-9b+a^{2}+3ab-2ba-6b^{2}.

$$\frac{\frac{2a^{2}-6b^{2}+7ab-9b}{a+3b}}{4a+2b}$$

Express \frac{\frac{2a^{2}-6b^{2}+7ab-9b}{a+3b}}{4a+2b} as a single fraction.

$$\frac{2a^{2}-6b^{2}+7ab-9b}{\left(a+3b\right)\left(4a+2b\right)}$$

Use the distributive property to multiply a+3b by 4a+2b and combine like terms.

$$\frac{2a^{2}-6b^{2}+7ab-9b}{4a^{2}+14ab+6b^{2}}$$