How to solve for a line intersecting a parabola but you don't know the gradient of the line

Question

Find the set of values of m for which the line $y = mx + 4$ intersects the curve $y = 3x^2 -4x +7$ at two distinct points.

That's the question and I'm really struggling to figure out how to solve it. Please help

Answer 1

We solve $$mx+4=3x^2-4x+7\implies3x^2-(4+m)x+3=0$$ The discriminant is thus $$b^2-4ac=(4+m)^2-36>0$$ for two distinct roots. This means that $$(4+m)^2>36\implies \cdots$$ Can you finish?

Answer 2

solve the equation $$mx+4=3x^2-4x+7$$ for $x$ and solve the equation $$m^2+8m-20=0$$ for $m$