Determine the condition to be satisfied by k such that the expression 2x^2+6x+1+k(x^2+2) is positive for all x belongs to real numbers .

Let 2x^2+6x+1+k(x^2+2)=0

(2+k)x^2+6x+1+2k=0

The graph must be a 'floating' graph on the positive side . I am not sure if i can say it that way. Thus , it has no real roots ..

6^2-4(2+k)(1+2k)<0

-8k^2-20k+24<0

I got the wrong range of values for k . Where is my mistake ?