Solve (a+b)(a-2b)+(4ab^3-8a^2b^2)div(-4ab)

$$( a + b ) ( a - 2 b ) + ( 4 a b ^ { 3 } - 8 a ^ { 2 } b ^ { 2 } ) \div ( - 4 a b )$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{4ab^{3}-8a^{2}b^{2}}{-4ab}.

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

Cancel out 4ab in both numerator and denominator.

$$\left(a+b\right)\left(a-2b\right)+\frac{b\left(-2a+b\right)}{-1}$$

Anything divided by -1 gives its opposite.

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

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

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

Use the distributive property to multiply b by -2a+b.

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

To find the opposite of -2ba+b^{2}, find the opposite of each term.

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

Combine -ab and 2ba to get ab.

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

Combine -2b^{2} and -b^{2} to get -3b^{2}.

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{4ab^{3}-8a^{2}b^{2}}{-4ab}.

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

Cancel out 4ab in both numerator and denominator.

$$\left(a+b\right)\left(a-2b\right)+\frac{b\left(-2a+b\right)}{-1}$$

Anything divided by -1 gives its opposite.

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

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

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

Use the distributive property to multiply b by -2a+b.

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

To find the opposite of -2ba+b^{2}, find the opposite of each term.

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

Combine -ab and 2ba to get ab.

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

Combine -2b^{2} and -b^{2} to get -3b^{2}.

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