The slope varies according to the derivative, f'(x)=12x-5. At f'(-2) the slope is 12(-2)-5=-29. At f'(-3) it is 12(-3)-5=-41. So the slope varies from point to point. If you need to find the slope between two points, the best you can get is the average, so in this case we would have the average of -29 and -41=-70/2=-35. Negative means a backwards slope (\).
Using h as the difference between two values of x we have: f(x)=6x^2-5x+4 and f(x+h)=6(x+h)^2-5(x+h)+4. If we calculate f(x+h)-f(x) we get: 6((x+h)^2-x^2)-5h=6(x+h-x)(x+h+x)-5h=6h(2x+h)-5h=12hx+6h^2-5h. However, h is supposed to be very small so that h^2 can be ignored. If h is 1, for example, it's too big to be ignored so that when we divide through by h, we get 12x+6h-5 instead of 12x-5, the derivative as shown above. Instead of -29 (for x=-2) we would have -29+6h. If h=-1, this becomes -35, which is the average slope.