jmc

The discriminant of the quadratic is $3^2-4(-12)(-5)=-231,$ which is negative. Therefore, the quadratic $-12x^2+3x-5=0$ has no real roots.

Furthermore, the coefficient of $x^2$ is $-12,$ which means that parabola is downward facing. Therefore, the inequalities is satisfied for all real numbers $x\in \boxed{(-\infty ,\infty )}.$