Solve 4x^3+6x^2-6x-4

$$4 x ^ { 3 } + 6 x ^ { 2 } - 6 x - 4$$

Factor

$$2\left(x-1\right)\left(x+2\right)\left(2x+1\right)$$

Solution Steps

Factor out 2.

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

Consider 2x^{3}+3x^{2}-3x-2. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -2 and q divides the leading coefficient 2. One such root is 1. Factor the polynomial by dividing it by x-1.

$$\left(x-1\right)\left(2x^{2}+5x+2\right)$$

Consider 2x^{2}+5x+2. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=5 ab=2\times 2=4$$

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 4.

$$1,4 2,2$$

Calculate the sum for each pair.

$$1+4=5 2+2=4$$

The solution is the pair that gives sum 5.

$$a=1 b=4$$

Rewrite 2x^{2}+5x+2 as \left(2x^{2}+x\right)+\left(4x+2\right).

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

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

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

Factor out common term 2x+1 by using distributive property.

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

Rewrite the complete factored expression.

$$2\left(x-1\right)\left(2x+1\right)\left(x+2\right)$$

Evaluate

$$2\left(x-1\right)\left(x+2\right)\left(2x+1\right)$$