Solve 9x^3+24x^2+12x

$$9 x ^ { 3 } + 24 x ^ { 2 } + 12 x$$

Factor

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

Solution Steps

Factor out 3.

$$3\left(3x^{3}+8x^{2}+4x\right)$$

Consider 3x^{3}+8x^{2}+4x. Factor out x.

$$x\left(3x^{2}+8x+4\right)$$

Consider 3x^{2}+8x+4. Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=8 ab=3\times 4=12$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. 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 8.

$$a=2 b=6$$

Rewrite 3x^{2}+8x+4 as \left(3x^{2}+2x\right)+\left(6x+4\right).

$$\left(3x^{2}+2x\right)+\left(6x+4\right)$$

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

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

Factor out common term 3x+2 by using distributive property.

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

Rewrite the complete factored expression.

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

Evaluate

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