Solve 12x^3+11x^2+2x

$$12 x ^ { 3 } + 11 x ^ { 2 } + 2 x$$

Factor

$$x\left(3x+2\right)\left(4x+1\right)$$

Solution Steps

Factor out x.

$$x\left(12x^{2}+11x+2\right)$$

Consider 12x^{2}+11x+2. Factor the expression by grouping. First, the expression needs to be rewritten as 12x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=11 ab=12\times 2=24$$

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 24.

$$1,24 2,12 3,8 4,6$$

Calculate the sum for each pair.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

The solution is the pair that gives sum 11.

$$a=3 b=8$$

Rewrite 12x^{2}+11x+2 as \left(12x^{2}+3x\right)+\left(8x+2\right).

$$\left(12x^{2}+3x\right)+\left(8x+2\right)$$

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

$$3x\left(4x+1\right)+2\left(4x+1\right)$$

Factor out common term 4x+1 by using distributive property.

$$\left(4x+1\right)\left(3x+2\right)$$

Rewrite the complete factored expression.

$$x\left(4x+1\right)\left(3x+2\right)$$

Evaluate

$$x\left(3x+2\right)\left(4x+1\right)$$