Solve f(x)=-2x^3-22x^2-68x-48

$$f ( x ) = - 2 x ^ { 3 } - 22 x ^ { 2 } - 68 x - 48$$

Factor

$$-2\left(x+1\right)\left(x+4\right)\left(x+6\right)$$

Solution Steps

Factor out 2.

$$2\left(-x^{3}-11x^{2}-34x-24\right)$$

Consider -x^{3}-11x^{2}-34x-24. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -24 and q divides the leading coefficient -1. One such root is -6. Factor the polynomial by dividing it by x+6.

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

Consider -x^{2}-5x-4. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-4. To find a and b, set up a system to be solved.

$$a+b=-5 ab=-\left(-4\right)=4$$

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 4.

$$-1,-4 -2,-2$$

Calculate the sum for each pair.

$$-1-4=-5 -2-2=-4$$

The solution is the pair that gives sum -5.

$$a=-1 b=-4$$

Rewrite -x^{2}-5x-4 as \left(-x^{2}-x\right)+\left(-4x-4\right).

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

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

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

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

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

Rewrite the complete factored expression.

$$2\left(x+6\right)\left(-x-1\right)\left(x+4\right)$$

Evaluate

$$-2\left(x+1\right)\left(x+4\right)\left(x+6\right)$$