Solve 3x^3+14x^2+8x

$$3 x ^ { 3 } + 14 x ^ { 2 } + 8 x$$

Factor

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

Solution Steps

Factor out x.

$$x\left(3x^{2}+14x+8\right)$$

Consider 3x^{2}+14x+8. Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx+8. To find a and b, set up a system to be solved.

$$a+b=14 ab=3\times 8=24$$

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 14.

$$a=2 b=12$$

Rewrite 3x^{2}+14x+8 as \left(3x^{2}+2x\right)+\left(12x+8\right).

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

Factor out x in the first and 4 in the second group.

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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