(A) Use interval notation to indicate the time interval or union of time
intervals when the particle is moving forward and backward

the particle is
speeding up

if  a(t) >0, it means 12t-48>0, and t
>4,  I = ]4, infinity[

the particle is
speeding down

if  a(t) < 0, it means 12t-48>0, and t <
4,  I = ] -infinity, 4[

 

when the particle is moving forward and backward

that is depending of the sign of v(t)

moving forward if
v(t)> 0

moving backward if
v(t)<  0

but v(t) = 6t² – 48 t
+90, this quadratic equation has exactly two solutions, t =1, and t =3

so  after taking care the sign of 6t² – 48 t +90,

 

v(t)> o when  t is in I = ] –inf, 1[ U ]4, inf[,the particle
is moving forward

v(t) < o
when  t is in I = ]1, 4[,the particle is
moving backward