Explanation

(d)
Given, $4\left( \sin^{2}{x - 3\sin x} \right) + 7$
$= 4\left\lbrack \left( \sin x - \frac{3}{2} \right)^{2} - \frac{9}{4} \right\rbrack + 7$
$= 4\left( \sin x - \frac{3}{2} \right)^{2} - 2$
Now, $- 1 \leq \sin x \leq 1$
$\Rightarrow \ - \frac{5}{2} \leq \sin{x - \frac{3}{2}} \leq - \frac{1}{2}$
$\Rightarrow \ \frac{1}{4} \leq \left( \sin{x - \frac{3}{2}} \right)^{2} \leq \frac{25}{4}$
$\Rightarrow \ \ 1 \leq 4\left( \sin x - \frac{3}{2} \right)^{2} \leq 25$
$\Rightarrow \mathbf{\ } - 1 \leq 4\ \left( \sin{x - \frac{3}{2}} \right)^{2} - 2 \leq 23$