Answer
$$s\left( t \right) = 2{t^2} - 3t + 2$$
Work Step by Step
$$\eqalign{
& a\left( t \right) = 4;{\text{ }}v\left( 0 \right) = - 3;{\text{ }}s\left( 0 \right) = 2 \cr
& v\left( t \right) = \int {a\left( t \right)dt} \cr
& v\left( t \right) = \int {4dt} \cr
& v\left( t \right) = 4t + C \cr
& {\text{Use the initial condition }}v\left( 0 \right) = - 3 \cr
& - 3 = 4\left( 0 \right) + C \cr
& C = - 3 \cr
& {\text{Thus,}} \cr
& v\left( t \right) = 4t - 3 \cr
& s\left( t \right) = \int {v\left( t \right)} dt \cr
& s\left( t \right) = \int {\left( {4t - 3} \right)} dt \cr
& s\left( t \right) = 2{t^2} - 3t + C \cr
& {\text{Use the initial condition }}s\left( 0 \right) = 2 \cr
& 2 = 2{\left( 0 \right)^2} - 3\left( 0 \right) + C \cr
& C = 2 \cr
& {\text{Therefore,}} \cr
& s\left( t \right) = 2{t^2} - 3t + 2 \cr} $$