The graph is a parabola with a minimum point.

So, Substitute the minimum value in the equation y = 3x^2 + 7x + m.

Thus the graph has two x-intercepts the value at the minimum point must be less than 0.

The graph is a parabola with the minimum where x= -b/2a = -7/6.
i.e

3(49/36) – 49/6 + m < 0

49/12 – 49/6 + m < 0

(49-98)/12 + m < 0

-49/12 + m < 0

m < 49/12