Do y=3x^2+5x+12 and y=3x-8 intercept with each other? If yes, at which point(s)?

This is like solving 2 simultaneous equations. Both equations are in a form of y=ax^2+bx+c. Equate both equations together and you got:
3x²+5x+12 = 3x-8

Put both polynomials at one side and you got:
3x²+2x+20 = 0

b²-4ac = 2² - 4(3)(20) = -236 <0
Because b² - 4ac <0, there are no real roots and the two equations do not intercept with each other.