Solve [2ab(-a+b)+6a^2b]:(-2ab)+[2a(a-b)-6ab]:[ax+a(1-x)]

$$[ 2 a b ( - a + b ) + 6 a ^ { 2 } b ] : ( - 2 a b ) + [ 2 a ( a - b ) - 6 a b ] : [ a x + a ( 1 - x ) ]$$

Evaluate

$$-9b$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{2ab\left(-a+b\right)+6a^{2}b}{-2ab}.

$$\frac{2ab\left(2a+b\right)}{-2ab}+\frac{2a\left(a-b\right)-6ab}{ax+a\left(1-x\right)}$$

Cancel out 2ab in both numerator and denominator.

$$\frac{2a+b}{-1}+\frac{2a\left(a-b\right)-6ab}{ax+a\left(1-x\right)}$$

Anything divided by -1 gives its opposite. To find the opposite of 2a+b, find the opposite of each term.

$$-2a-b+\frac{2a\left(a-b\right)-6ab}{ax+a\left(1-x\right)}$$

Factor ax+a\left(1-x\right).

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

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

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

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

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

Do the multiplications in \left(-2a-b\right)a+2a\left(a-b\right)-6ab.

$$\frac{-2a^{2}-ba+2a^{2}-2ab-6ab}{a}$$

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

$$\frac{-9ba}{a}$$

Cancel out a in both numerator and denominator.

$$-9b$$

Expand

$$-9b$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{2ab\left(-a+b\right)+6a^{2}b}{-2ab}.

$$\frac{2ab\left(2a+b\right)}{-2ab}+\frac{2a\left(a-b\right)-6ab}{ax+a\left(1-x\right)}$$

Cancel out 2ab in both numerator and denominator.

$$\frac{2a+b}{-1}+\frac{2a\left(a-b\right)-6ab}{ax+a\left(1-x\right)}$$

Anything divided by -1 gives its opposite. To find the opposite of 2a+b, find the opposite of each term.

$$-2a-b+\frac{2a\left(a-b\right)-6ab}{ax+a\left(1-x\right)}$$

Factor ax+a\left(1-x\right).

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

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

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

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

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

Do the multiplications in \left(-2a-b\right)a+2a\left(a-b\right)-6ab.

$$\frac{-2a^{2}-ba+2a^{2}-2ab-6ab}{a}$$

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

$$\frac{-9ba}{a}$$

Cancel out a in both numerator and denominator.

$$-9b$$