Solve {l}{x^6-1}{(x+4)^3(x-2)+(x+4)^2(x-2)^2}

$$\left. \begin{array} { l } { x ^ { 6 } - 1 } \\ { ( x + 4 ) ^ { 3 } ( x - 2 ) + ( x + 4 ) ^ { 2 } ( x - 2 ) ^ { 2 } } \end{array} \right.$$

Least Common Multiple

$$2\left(x-2\right)\left(x^{2}-1\right)\left(x+4\right)^{2}\left(\left(x^{2}+1\right)^{2}-x^{2}\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$\left(x^{4}+x^{2}+1\right)\left(x^{2}-1\right)=\left(x-1\right)\left(x+1\right)\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)$$

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

$$2\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+4\right)^{2}\left(x^{2}-x+1\right)\left(x^{2}+x+1\right)$$

Expand the expression.

$$2x^{9}+12x^{8}-64x^{6}-2x^{3}-12x^{2}+64$$

Evaluate

$$x^{6}-1,\ 2\left(x-2\right)\left(x+1\right)\left(x+4\right)^{2}$$