Acceleration can be found by differentiating v(t) by t:
dv/dt =d(t^2+2)/dt
=d(t^2)/dt + d(2)/dt
=2tdt/dt + 0
=2t
The displacement x(t) can be found by integrating v(t) by t
x(t) = int(t^2 + 2)dt
= t^3/3 + 2t + c
When t=0, x(0)=4=0+0+c
So c=4
x(t) = t^3/3 + 2t + 4
v(t) = t^2 + 2
a(t) = 2t