Solve {l}{9a^{2}-12ab+15a^{3}b^{2}-24ab^{2}}{6x^{3}y^2-8x^2y-24x^4y^4-40}

$$\left. \begin{array} { l } { 9 a ^ { 2 } - 12 a b + 15 a ^ { 3 } b ^ { 2 } - 24 a b ^ { 2 } } \\ { 6 x ^ { 3 } y ^ { 2 } - 8 x ^ { 2 } y - 24 x ^ { 4 } y ^ { 4 } - 40 } \end{array} \right.$$

Least Common Multiple

$$6a\left(12by^{2}x^{3}-9ay^{2}x^{3}-15x^{3}\left(aby\right)^{2}+24x^{3}\left(by\right)^{2}+12ayx^{2}-16byx^{2}+20y\left(abx\right)^{2}-32y\left(bx\right)^{2}+60\left(ab\right)^{2}\left(xy\right)^{4}-96b^{2}\left(xy\right)^{4}+36a\left(xy\right)^{4}-48b\left(xy\right)^{4}+100\left(ab\right)^{2}-160b^{2}+60a-80b\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$6y^{2}x^{3}-8yx^{2}-24y^{4}x^{4}-40=2\left(-12x^{4}y^{4}+3y^{2}x^{3}-4yx^{2}-20\right)$$

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

$$6a\left(60a^{2}b^{2}x^{4}y^{4}-96b^{2}x^{4}y^{4}+36ax^{4}y^{4}-48bx^{4}y^{4}-15a^{2}b^{2}y^{2}x^{3}+24b^{2}y^{2}x^{3}+12by^{2}x^{3}-9ay^{2}x^{3}+20ya^{2}b^{2}x^{2}-32yb^{2}x^{2}+12ayx^{2}-16byx^{2}+100a^{2}b^{2}-160b^{2}+60a-80b\right)$$

Expand the expression.

$$360b^{2}a^{3}x^{4}y^{4}-576ab^{2}x^{4}y^{4}+216a^{2}x^{4}y^{4}-288abx^{4}y^{4}-90b^{2}y^{2}a^{3}x^{3}+144ab^{2}y^{2}x^{3}+72aby^{2}x^{3}-54a^{2}y^{2}x^{3}+120yb^{2}x^{2}a^{3}-192ayb^{2}x^{2}+72ya^{2}x^{2}-96abyx^{2}+600b^{2}a^{3}-960ab^{2}-480ab+360a^{2}$$

Evaluate

$$3a\left(5\left(ab\right)^{2}-8b^{2}+3a-4b\right),\ 2\left(3y^{2}x^{3}-4yx^{2}-12\left(xy\right)^{4}-20\right)$$