Solve {l}{2(a-b)^{2}-3(b-a)-2(b-a)^{2}+4(a-b)-(a-b)}{2b^{2}-8b-3a-2b^2+2a^2+4a-4b-a^3+b^3}

$$\left. \begin{array} { l } { 2 ( a - b ) ^ { 2 } - 3 ( b - a ) - 2 ( b - a ) ^ { 2 } + 4 ( a - b ) - ( a - b ) } \\ { 2 b ^ { 2 } - 8 b - 3 a - 2 b ^ { 2 } + 2 a ^ { 2 } + 4 a - 4 b - a ^ { 3 } + b ^ { 3 } } \end{array} \right.$$

Least Common Multiple

$$6\left(a-b\right)\left(a^{3}-2a^{2}-a-b^{3}+12b\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$6a-6b=6\left(a-b\right)$$

Identify all the factors and their highest power in all expressions. Multiply the highest powers of these factors to get the least common multiple.

$$6\left(a-b\right)\left(a^{3}-2a^{2}-a-b^{3}+12b\right)$$

Expand the expression.

$$6a^{4}-12a^{3}-6a^{2}-6ab^{3}+78ab+6b^{4}-72b^{2}-6ba^{3}+12ba^{2}$$

Evaluate

$$6\left(a-b\right),\ -a^{3}+2a^{2}+a+b^{3}-12b$$