Solve -2x^5-14x^4-12x^3

$$-2 { x }^{ 5 } -14 { x }^{ 4 } -12 { x }^{ 3 }$$

Factor

$$-2\left(x+1\right)\left(x+6\right)x^{3}$$

Solution Steps

Factor out 2.

$$2\left(-x^{5}-7x^{4}-6x^{3}\right)$$

Consider -x^{5}-7x^{4}-6x^{3}. Factor out x^{3}.

$$x^{3}\left(-x^{2}-7x-6\right)$$

Consider -x^{2}-7x-6. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=-7 ab=-\left(-6\right)=6$$

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 6.

$$-1,-6 -2,-3$$

Calculate the sum for each pair.

$$-1-6=-7 -2-3=-5$$

The solution is the pair that gives sum -7.

$$a=-1 b=-6$$

Rewrite -x^{2}-7x-6 as \left(-x^{2}-x\right)+\left(-6x-6\right).

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

Factor out x in the first and 6 in the second group.

$$x\left(-x-1\right)+6\left(-x-1\right)$$

Factor out common term -x-1 by using distributive property.

$$\left(-x-1\right)\left(x+6\right)$$

Rewrite the complete factored expression.

$$2x^{3}\left(-x-1\right)\left(x+6\right)$$

Evaluate

$$-2\left(x+1\right)\left(x+6\right)x^{3}$$