Solve 6x^3+8x^2-8x

$$6 { x }^{ 3 } +8 { x }^{ 2 } -8x$$

Factor

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

Solution Steps

Factor out 2.

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

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

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

Consider 3x^{2}+4x-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.

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

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -12.

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

Calculate the sum for each pair.

$$-1+12=11 -2+6=4 -3+4=1$$

The solution is the pair that gives sum 4.

$$a=-2 b=6$$

Rewrite 3x^{2}+4x-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.

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

Evaluate

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