I'll take a shot at this...
1. Differentiate the equation to find the x,y coordinates where the slope=2: The derivative is 6x+6, which is equal to 2 where X = -2/3. Plugging -2/3 into your original equation gives us Y = -11/3.
2. If we want the line Y=2x+b to be tangential to this (i.e., touch it only once), we'd need to solve for b using the above x,y pairing:
-11/3 = 2* (-2/3) + b. If I did the math correctly, b = -7/3.
3. So if the y intercept is -7/3, we intersect your equation at only one point. If the y intercept is less than -7/3 (i.e., more negative), we wouldn't intersect your equation at all. So I believe the answer is that as long as the b in Y=2x+b is strictly greater than -7/3, we ensure that we cross through the line your equation represents exactly twice.
- Steve J