First off, the leading coefficient must be positive in order for the expression to be positive for all x belonging toR, so

2 + k > 0 =>k > -2.

The last inequality should beThe 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 ?

So, divide both sides by -4:

The solution set for k is

.

But I said earlier thatk must be greater than -2, so intersect this with the solution set and you get

.

01