Answer
$$40t + 8$$
Work Step by Step
$$\eqalign{
& {\bf{r}}\left( t \right) = \left( {{t^2} + 4t} \right){\bf{i}} - 3{t^2}{\bf{j}} \cr
& \left( {\text{a}} \right){\text{Calculate }}{\bf{r}}{\text{'}}\left( t \right) \cr
& {\bf{r}}{\text{'}}\left( t \right) = \frac{d}{{dt}}\left[ {{t^2} + 4t} \right]{\bf{i}} - \frac{d}{{dt}}\left[ {3{t^2}} \right]{\bf{j}} \cr
& {\bf{r}}{\text{'}}\left( t \right) = \left( {2t + 4} \right){\bf{i}} - 6t{\bf{j}} \cr
& \cr
& \left( {\text{b}} \right){\text{Calculate }}{\bf{r}}{\text{''}}\left( t \right) \cr
& {\bf{r}}{\text{''}}\left( t \right) = \frac{d}{{dt}}\left[ {{\bf{r}}{\text{'}}\left( t \right)} \right] \cr
& {\bf{r}}{\text{''}}\left( t \right) = \frac{d}{{dt}}\left[ {2t + 4} \right]{\bf{i}} + \frac{d}{{dt}}\left[ { - 6t} \right]{\bf{j}} \cr
& {\bf{r}}{\text{''}}\left( t \right) = 2{\bf{i}} - 6{\bf{j}} \cr
& \cr
& \left( {\text{c}} \right){\text{Calculate the dot product }}{\bf{r}}{\text{'}}\left( t \right) \cdot {\bf{r}}{\text{''}}\left( t \right) \cr
& {\bf{r}}{\text{'}}\left( t \right) \cdot {\bf{r}}{\text{''}}\left( t \right) = \left[ {\left( {2t + 4} \right){\bf{i}} - 6t{\bf{j}}} \right] \cdot \left( {2{\bf{i}} - 6{\bf{j}}} \right) \cr
& {\bf{r}}{\text{'}}\left( t \right) \cdot {\bf{r}}{\text{''}}\left( t \right) = \left( {2t + 4} \right)\left( 2 \right) + \left( { - 6t} \right)\left( { - 6} \right) \cr
& {\bf{r}}{\text{'}}\left( t \right) \cdot {\bf{r}}{\text{''}}\left( t \right) = 4t + 8 + 36t \cr
& {\bf{r}}{\text{'}}\left( t \right) \cdot {\bf{r}}{\text{''}}\left( t \right) = 40t + 8 \cr} $$
