Solve -12b^3-12b^2-3b=

$$- 12 b ^ { 3 } - 12 b ^ { 2 } - 3 b =$$

Factor

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

Solution Steps

Factor out 3.

$$3\left(-4b^{3}-4b^{2}-b\right)$$

Consider -4b^{3}-4b^{2}-b. Factor out b.

$$b\left(-4b^{2}-4b-1\right)$$

Consider -4b^{2}-4b-1. Factor the expression by grouping. First, the expression needs to be rewritten as -4b^{2}+pb+qb-1. To find p and q, set up a system to be solved.

$$p+q=-4 pq=-4\left(-1\right)=4$$

Since pq is positive, p and q have the same sign. Since p+q is negative, p and q 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 -4.

$$p=-2 q=-2$$

Rewrite -4b^{2}-4b-1 as \left(-4b^{2}-2b\right)+\left(-2b-1\right).

$$\left(-4b^{2}-2b\right)+\left(-2b-1\right)$$

Factor out -2b in the first and -1 in the second group.

$$-2b\left(2b+1\right)-\left(2b+1\right)$$

Factor out common term 2b+1 by using distributive property.

$$\left(2b+1\right)\left(-2b-1\right)$$

Rewrite the complete factored expression.

$$3b\left(2b+1\right)\left(-2b-1\right)$$

Evaluate

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