Solve -x^3-12x^2-32x

$$- x ^ { 3 } - 12 x ^ { 2 } - 32 x$$

Factor

$$-x\left(x+4\right)\left(x+8\right)$$

Solution Steps

Factor out x.

$$x\left(-x^{2}-12x-32\right)$$

Consider -x^{2}-12x-32. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-32. To find a and b, set up a system to be solved.

$$a+b=-12 ab=-\left(-32\right)=32$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 32.

$$-1,-32 -2,-16 -4,-8$$

Calculate the sum for each pair.

$$-1-32=-33 -2-16=-18 -4-8=-12$$

The solution is the pair that gives sum -12.

$$a=-4 b=-8$$

Rewrite -x^{2}-12x-32 as \left(-x^{2}-4x\right)+\left(-8x-32\right).

$$\left(-x^{2}-4x\right)+\left(-8x-32\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(-x-4\right)+8\left(-x-4\right)$$

Factor out common term -x-4 by using distributive property.

$$\left(-x-4\right)\left(x+8\right)$$

Rewrite the complete factored expression.

$$x\left(-x-4\right)\left(x+8\right)$$

Evaluate

$$-x\left(x+4\right)\left(x+8\right)$$