2x^4+x^3+6x+3> 0 solve the system of inequalities

Solve for where it equals 0
use grouping

(2x^4+x^3)+(6x+3)=0
(x^3)(2x+1)+(3)(2x+1)=0
(x^3+3)(2x+1)=0
set equal to zero

x^3+3=0
x^3=-3
x=-∛3

2x+1=0
2x=-1
x=-1/2

now see where the inequality is true

evaluate for x=0
we get 3>0, true
so it it outside the range from -∛3 to -1/2

basically

x<-∛3 or x>-1/2
those are the solutionsets