Solve -1/5x^3-8/5x^2-16/5x

$$- \frac{ 1 }{ 5 } { x }^{ 3 } - \frac{ 8 }{ 5 } { x }^{ 2 } - \frac{ 16 }{ 5 } x$$

Factor

$$-\frac{x\left(x+4\right)^{2}}{5}$$

Solution Steps

Factor out \frac{1}{5}.

$$\frac{-x^{3}-8x^{2}-16x}{5}$$

Consider -x^{3}-8x^{2}-16x. Factor out x.

$$x\left(-x^{2}-8x-16\right)$$

Consider -x^{2}-8x-16. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-16. To find a and b, set up a system to be solved.

$$a+b=-8 ab=-\left(-16\right)=16$$

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 16.

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

Calculate the sum for each pair.

$$-1-16=-17 -2-8=-10 -4-4=-8$$

The solution is the pair that gives sum -8.

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

Rewrite -x^{2}-8x-16 as \left(-x^{2}-4x\right)+\left(-4x-16\right).

$$\left(-x^{2}-4x\right)+\left(-4x-16\right)$$

Factor out -x in the first and -4 in the second group.

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

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

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

Rewrite the complete factored expression.

$$\frac{x\left(x+4\right)\left(-x-4\right)}{5}$$

Evaluate

$$-\frac{x\left(x+4\right)^{2}}{5}$$