Solve {l}{x(2x^{2}+3xy+y^{2})-y(5x^{2}-2xy+y^{2})}{a^{2}b(a^3+5ab-b^3)+2ab(a^2-2a^2b)}

$$\left. \begin{array} { l } { x ( 2 x ^ { 2 } + 3 x y + y ^ { 2 } ) - y ( 5 x ^ { 2 } - 2 x y + y ^ { 2 } ) } \\ { a ^ { 2 } b ( a ^ { 3 } + 5 a b - b ^ { 3 } ) + 2 a b ( a ^ { 2 } - 2 a ^ { 2 } b ) } \end{array} \right.$$

Least Common Multiple

$$ba^{2}\left(2abx^{3}+4ax^{3}+3xy^{2}a^{3}-3xy^{2}b^{3}+3abxy^{2}+6axy^{2}-aby^{3}-2ay^{3}+2yx^{2}b^{3}-2yx^{2}a^{3}-2abyx^{2}-4ayx^{2}+2\left(ax\right)^{3}+\left(by\right)^{3}-2\left(bx\right)^{3}-\left(ay\right)^{3}\right)$$

Solution Steps

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

$$ba^{2}\left(2a^{3}x^{3}-2b^{3}x^{3}+2abx^{3}+4ax^{3}+3xy^{2}a^{3}-3xy^{2}b^{3}+3abxy^{2}+6axy^{2}+b^{3}y^{3}-a^{3}y^{3}-aby^{3}-2ay^{3}+2yx^{2}b^{3}-2yx^{2}a^{3}-2abyx^{2}-4ayx^{2}\right)$$

Expand the expression.

$$2bx^{3}a^{5}-2a^{2}x^{3}b^{4}+2b^{2}a^{3}x^{3}+4ba^{3}x^{3}+3bxy^{2}a^{5}-3xa^{2}y^{2}b^{4}+3xb^{2}y^{2}a^{3}+6bxy^{2}a^{3}+a^{2}y^{3}b^{4}-by^{3}a^{5}-b^{2}a^{3}y^{3}-2ba^{3}y^{3}+2ya^{2}x^{2}b^{4}-2byx^{2}a^{5}-2yb^{2}x^{2}a^{3}-4byx^{2}a^{3}$$

Evaluate

$$2x^{3}+3xy^{2}-y^{3}-2yx^{2},\ ba^{2}\left(ab+a^{3}-b^{3}+2a\right)$$