please help me work through this! I'm just a bit stuck and think help would make me better understand

Answer:
Given equation of motion of a particle is,
[tex]s(t)=4t^3+36t^2+96t[/tex]To find the velocity function of t, value of t when the velocity is zero and acceleration of the particle.
Explanation:
we know that, velocity of a particle is rate of change of the object’s position with respect to a frame of reference and time.
we get,
[tex]v(t)=\frac{d}{dt}(s(t))[/tex]Substitute for s(t), we get
[tex]v(t)=\frac{d}{dt}(4t^3+36t^2+96t)[/tex]Differenting we get,
[tex]v(t)=12t^2+72t+96[/tex]Required velocity function is,
[tex]v(t)=12t^2+72t+96[/tex]when velocity equal to 0, we get
[tex]12t^2+72t+96=0[/tex]Dividing by 12, we get
[tex]t^2+6t+8=0[/tex]we get,
[tex]t^2+4t+2t+8=0[/tex][tex]t(t+4)+2(t+4)=0[/tex]Taking (t+4) as common we get,
[tex](t+4)(t+2)=0[/tex]we get,
[tex]t=-4,-2[/tex]The values of t are,
t=-4 and t=-2
To find acceleration of a particle.
we know that,
Acceleration a(t) is the rate at which velocity (speed) is changing.
we get,
[tex]a(t)=\frac{d}{dt}(v(t))[/tex]Substitute v(t), we get,
[tex]a(t)=\frac{d}{dt}(12t^2+72t+96)[/tex]we get,
[tex]a(t)=24t+72[/tex]The acceleration of a particle is,
[tex]a(t)=24t+72[/tex]