Solve x(x-1)>(x+1)(3x+2)-2>0

$$x(x-1) > (x+1)(3x+2)-2 > 0$$

Solve for x

$$x\in \left(-\infty,-\frac{5}{3}\right)\cup \left(0,\infty\right)$$

Solution Steps

Use the distributive property to multiply x+1 by 3x+2 and combine like terms.

$$3x^{2}+5x+2-2>0$$

Subtract 2 from 2 to get 0.

$$3x^{2}+5x>0$$

Factor out x.

$$x\left(3x+5\right)>0$$

For the product to be positive, x+\frac{5}{3} and x have to be both negative or both positive. Consider the case when x+\frac{5}{3} and x are both negative.

$$x+\frac{5}{3}<0 x<0$$

The solution satisfying both inequalities is x<-\frac{5}{3}.

$$x<-\frac{5}{3}$$

Consider the case when x+\frac{5}{3} and x are both positive.

$$x>0 x+\frac{5}{3}>0$$

The solution satisfying both inequalities is x>0.

$$x>0$$

The final solution is the union of the obtained solutions.

$$x<-\frac{5}{3}\text{; }x>0$$