Answer
$$\left( {\bf{a}} \right)v\left( t \right) = 12{t^2} + 16t + 1 ,\,\,\,\,\left( {\bf{b}} \right)v\left( 0 \right) = 1;\,\,\,\,\,\,\,\,v\left( 5 \right) = 381;\,\,\,\,\,\,v\left( {10} \right) = 1361$$
Work Step by Step
$$\eqalign{
& {\text{let }}s\left( t \right) = 4{t^3} + 8{t^2} + t \cr
& \left( {\bf{a}} \right){\text{find the velocity using }}v\left( t \right) = s'\left( t \right).{\text{ then}}{\text{,}} \cr
& v\left( t \right) = s'\left( t \right) = {D_t}\left( {4{t^3} + 8{t^2} + t} \right) \cr
& {\text{solve the derivatives using the power rule}} \cr
& v\left( t \right) = 4\left( {3{t^2}} \right) + 8\left( {2t} \right) + 1 \cr
& v\left( t \right) = 12{t^2} + 16t + 1 \cr
& \cr
& \left( {\bf{b}} \right){\text{ evaluate the velocity }}v\left( t \right){\text{ at }}t = 0,{\text{ }}t = 5{\text{ and }}t = 10 \cr
& v\left( 0 \right) = 12{\left( 0 \right)^2} + 16\left( 0 \right) + 1 = 1 \cr
& v\left( 5 \right) = 12{\left( 5 \right)^2} + 16\left( 5 \right) + 1 = 381 \cr
& v\left( {10} \right) = 12{\left( {10} \right)^2} + 16\left( {10} \right) + 1 = 1361 \cr} $$
