Solve 24x^3+18x^2-6x

$$24 x ^ { 3 } + 18 x ^ { 2 } - 6 x$$

Factor

$$6x\left(4x-1\right)\left(x+1\right)$$

Solution Steps

Factor out 6.

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

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

$$x\left(4x^{2}+3x-1\right)$$

Consider 4x^{2}+3x-1. Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=3 ab=4\left(-1\right)=-4$$

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 -4.

$$-1,4 -2,2$$

Calculate the sum for each pair.

$$-1+4=3 -2+2=0$$

The solution is the pair that gives sum 3.

$$a=-1 b=4$$

Rewrite 4x^{2}+3x-1 as \left(4x^{2}-x\right)+\left(4x-1\right).

$$\left(4x^{2}-x\right)+\left(4x-1\right)$$

Factor out x in 4x^{2}-x.

$$x\left(4x-1\right)+4x-1$$

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

$$\left(4x-1\right)\left(x+1\right)$$

Rewrite the complete factored expression.

$$6x\left(4x-1\right)\left(x+1\right)$$

Evaluate

$$6x\left(4x-1\right)\left(x+1\right)$$