Sinc that is a quadratic with positive leading coefficient, its graph is a parabola opening upward. If its vertex has negative y value then its value is negative for x between the zeros and postive for x outside the zeros.
R.M. has already told you that this factors into (2x- 3)(x- 4) so its zeros are x= 3/2 and x= -4. The inequality is satisfied for x< -4 and x> 3/2.
Or you could do what R.M. suggested, evaluate at a single point in each interval. The numbers x= -4 and x= 3/2 divide the number line into three intervals- x< -4, -4< x< 3/2, and x> 3/2. -5 is less than -4 and if x= -5 we have 2(-5)^2+ 5(-5)- 12= 50- 25- 12= 25- 12= 13> 0. 0 is between -4 and 3/2 and if x= 0 we have 2(0)^2+ 5(0)- 12= -12< 0. Finally, 2 is greater than 3/2 and if x= 2 we have 2(2)^3+ 5(2)- 12= 8+ 10- 12= 6> 0.
Yet another method- The product of two numbers is negative if and only if they have different sign. If they are either both positive or both negative then the product is positive.
2x- 3> 0 for x> 3/2 and x+ 4> 0 for x> -4. Those are both true if x> 3/2.
2x- 3< 0 for x< 3/2 and x+ 4< 0 for x< -4. Those are both true if x< -4.
Again, we have that 2x^2+ 5x- 12> 0 if x< -4 or if x> 3/2.