Solve {l}{(3a^{3}+5a^2-4)*(3a)}{(3x^2)(2x^3-3x^2+4x-2)}

$$\left. \begin{array} { l } { ( 3 a ^ { 3 } + 5 a ^ { 2 } - 4 ) \cdot ( 3 a ) } \\ { ( 3 x ^ { 2 } ) ( 2 x ^ { 3 } - 3 x ^ { 2 } + 4 x - 2 ) } \end{array} \right.$$

Least Common Multiple

$$3ax^{2}\left(10a^{2}x^{3}-8x^{3}-9x^{2}a^{3}+12x^{2}+12xa^{3}+20xa^{2}-16x+6\left(ax\right)^{3}-15\left(ax\right)^{2}-6a^{3}-10a^{2}+8\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$9a^{4}+15a^{3}-12a=3a\left(3a^{3}+5a^{2}-4\right) 6x^{5}-9x^{4}+12x^{3}-6x^{2}=3x^{2}\left(2x^{3}-3x^{2}+4x-2\right)$$

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

$$3ax^{2}\left(6a^{3}x^{3}+10a^{2}x^{3}-8x^{3}-9x^{2}a^{3}-15a^{2}x^{2}+12x^{2}+12xa^{3}+20xa^{2}-16x-6a^{3}-10a^{2}+8\right)$$

Expand the expression.

$$18a^{4}x^{5}+30a^{3}x^{5}-24ax^{5}-27a^{4}x^{4}-45a^{3}x^{4}+36ax^{4}+36x^{3}a^{4}+60a^{3}x^{3}-48ax^{3}-18x^{2}a^{4}-30x^{2}a^{3}+24ax^{2}$$

Evaluate

$$9a^{4}+15a^{3}-12a,\ 6x^{5}-9x^{4}+12x^{3}-6x^{2}$$