Solve {l}{-4x^{3}-4x^2+16x+16}{x^4-4x^3-x^2+16x-12}

$$\left. \begin{array} { l } { - 4 x ^ { 3 } - 4 x ^ { 2 } + 16 x + 16 } \\ { x ^ { 4 } - 4 x ^ { 3 } - x ^ { 2 } + 16 x - 12 } \end{array} \right.$$

Least Common Multiple

$$4\left(x-3\right)\left(x^{2}-4\right)\left(x^{2}-1\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$-4\left(1+x\right)\left(x^{2}-4\right)=-4\left(x-2\right)\left(x+1\right)\left(x+2\right) \left(-3+x\right)\left(-1+x\right)\left(x^{2}-4\right)=\left(x-3\right)\left(x-2\right)\left(x-1\right)\left(x+2\right)$$

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

$$4\left(x-3\right)\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)$$

Expand the expression.

$$4x^{5}-12x^{4}-20x^{3}+60x^{2}+16x-48$$

Evaluate

$$-4\left(x+1\right)\left(x^{2}-4\right),\ \left(x-3\right)\left(x-1\right)\left(x^{2}-4\right)$$