When is the particle moving forward (in the positive direction)?

equation: v(t)= 3t^2-12t+9> 0

Factor that polynomial you got there, to =3(t-1)(t-3)- then you find that when t<1 and when t>3, your equation is positive. this function is stating that when v(t) is positive, the particle is moving in the "positive" direction and when the function is negative, it's moving in the "negative" direction.
v(t) = 3t^2 - 12t + 9
v(t) = 3 * (t^2 - 4t + 3)
v(t) = 3 * (t - 1)(t - 3)

It's when the quantities (t-1) and (t-3) are both either negative or positive. Because then you have 3 * a positive scalar quantity. So when t > 3, both (t-1) and (t-3) will be positive. When t < 1 both will be negative. It can't be when t>1 and t>3, because that would be the same as saying t>1. You only have one t, it can't be two things at once. Plus, think about when t=2. (t-2) is positive but (t-3) is negative. It will be traveling in the negative direction. Hope that helps clear things up a bit. :)

Liquid. good luck : )