Determine the restrictions on the y-intercept so that $\displaystyle y=3x^2+6x-1 $ intersects with a line with slope 2 in more then one place
let k = y-intercept for the line
points of intersection can be found by setting the linear and quadratic equations equal to each other ...
$\displaystyle 3x^2 + 6x - 1 = 2x + k$
set the equation equal to 0 ...
$\displaystyle 3x^2 + 4x - (1+k) = 0$
if the line intersects the parabola twice, then the above quadratic must have two distinct real roots.
for the quadratic to have two distinct real roots, the discriminant, $\displaystyle b^2-4ac > 0$
$\displaystyle 4^2 - 4(3)[-(1+k)] > 0$
solve the inequality for k.