Explanation

(a)
The general term in $\left( \frac{x}{2} - \frac{3}{x^{2}} \right)^{10}$ is
$T_{r + 1} = ( - 1)^{r}\ ^{10}C_{r}\left( \frac{x}{2} \right)^{10 - r}\left( \frac{3}{x^{2}} \right)^{r}$
$= ( - 1)^{r}\ ^{10}C_{r}.\frac{3^{r}}{2^{10 - r}}.x^{10 - 3r}$
For coefficient of $x^{4}$ , we have to take $10 - 3r = 4$
$\Rightarrow \ 3r = 6\ \Rightarrow \ r = 2$
$\therefore$ Coefficient of $x^{4}$ in $\left( \frac{x}{2} - \frac{3}{x^{2}} \right)^{10}$
$= ( - 1)^{2}.\ ^{10}C_{2}.\frac{3^{2}}{2^{8}} = \frac{9 \times 45}{256} = \frac{405}{256}$