Solve -12r^3-21r^2-9r

$$- 12 r ^ { 3 } - 21 r ^ { 2 } - 9 r$$

Factor

$$-3r\left(r+1\right)\left(4r+3\right)$$

Solution Steps

Factor out 3.

$$3\left(-4r^{3}-7r^{2}-3r\right)$$

Consider -4r^{3}-7r^{2}-3r. Factor out r.

$$r\left(-4r^{2}-7r-3\right)$$

Consider -4r^{2}-7r-3. Factor the expression by grouping. First, the expression needs to be rewritten as -4r^{2}+ar+br-3. To find a and b, set up a system to be solved.

$$a+b=-7 ab=-4\left(-3\right)=12$$

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 12.

$$-1,-12 -2,-6 -3,-4$$

Calculate the sum for each pair.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

The solution is the pair that gives sum -7.

$$a=-3 b=-4$$

Rewrite -4r^{2}-7r-3 as \left(-4r^{2}-3r\right)+\left(-4r-3\right).

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

Factor out -r in the first and -1 in the second group.

$$-r\left(4r+3\right)-\left(4r+3\right)$$

Factor out common term 4r+3 by using distributive property.

$$\left(4r+3\right)\left(-r-1\right)$$

Rewrite the complete factored expression.

$$3r\left(4r+3\right)\left(-r-1\right)$$

Evaluate

$$-3r\left(r+1\right)\left(4r+3\right)$$