Answer
$$\left( {\bf{a}} \right) - 9{t^2} + 8t - 10,\,\,\,\,\left( {\bf{b}} \right)v\left( 0 \right) = - 10;\,\,\,\,\,\,\,\,v\left( 5 \right) = - 195;\,\,\,\,\,\,v\left( {10} \right) = - 830$$
Work Step by Step
$$\eqalign{
& {\text{let }}s\left( t \right) = - 3{t^3} + 4{t^2} - 10t + 5 \cr
& \left( {\bf{a}} \right){\text{find the velocity using }}v\left( t \right) = s'\left( t \right).{\text{ then}}{\text{,}} \cr
& v\left( t \right) = s'\left( t \right) = {D_t}\left( { - 3{t^3} + 4{t^2} - 10t + 5} \right) \cr
& {\text{solve the derivatives using the power rule}} \cr
& v\left( t \right) = - 3\left( {3{t^2}} \right) + 4\left( {2t} \right) - 10\left( 1 \right) + 0 \cr
& v\left( t \right) = - 9{t^2} + 8t - 10 \cr
& \cr
& \left( {\bf{b}} \right){\text{ evaluate the velocity }}v\left( t \right){\text{ at }}t = 0,{\text{ }}t = 5{\text{ and }}t = 10 \cr
& v\left( 0 \right) = - 9{\left( 0 \right)^2} + 8\left( 0 \right) - 10 = - 10 \cr
& v\left( 5 \right) = - 9{\left( 5 \right)^2} + 8\left( 5 \right) - 10 = - 195 \cr
& v\left( {10} \right) = - 9{\left( {10} \right)^2} + 8\left( {10} \right) - 10 = - 830 \cr} $$
