Given

\(\displaystyle{y}={3}{x}^{{2}}+{7}{x}+{m}\)

The graph of a quadratic function has two x-intecepts when the discriminant of the equation is positive.

The discriminant of a quadratic equation \(\displaystyle{y}={a}{x}^{{2}}+{b}{x}+{c}\ {i}{s}\ {D}={b}^{{2}}-{4}{a}{c}.\)

\(D=b^2-4ac =7^2-4(3)(m) =49-12m\)

The discriminant needs to be positive when the graph has two x-intecepts:

49-12m>0

Subtract 49 from each side of the inequality:

-12m>-49

Divide each side of the inequality by -12

\(\displaystyle{m}{<}{\left(-\frac{{49}}{{-{{12}}}}\right)}\)

Simplify:

\(\displaystyle{m}{<}\frac{{49}}{{12}}\)