Solve -6x^3-17x^2-12x

$$- 6 x ^ { 3 } - 17 x ^ { 2 } - 12 x$$

Factor

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

Solution Steps

Factor out x.

$$x\left(-6x^{2}-17x-12\right)$$

Consider -6x^{2}-17x-12. Factor the expression by grouping. First, the expression needs to be rewritten as -6x^{2}+ax+bx-12. To find a and b, set up a system to be solved.

$$a+b=-17 ab=-6\left(-12\right)=72$$

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 72.

$$-1,-72 -2,-36 -3,-24 -4,-18 -6,-12 -8,-9$$

Calculate the sum for each pair.

$$-1-72=-73 -2-36=-38 -3-24=-27 -4-18=-22 -6-12=-18 -8-9=-17$$

The solution is the pair that gives sum -17.

$$a=-8 b=-9$$

Rewrite -6x^{2}-17x-12 as \left(-6x^{2}-8x\right)+\left(-9x-12\right).

$$\left(-6x^{2}-8x\right)+\left(-9x-12\right)$$

Factor out 2x in the first and 3 in the second group.

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

Factor out common term -3x-4 by using distributive property.

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

Rewrite the complete factored expression.

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

Evaluate

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