Explanation

(b)
$\because\ x^{3} + x^{2} + 4x + 2\sin x = 0$
$\Rightarrow x^{3} + (x + 2)^{2} + 2\sin x = 4$
$x = 0$, satisfies this equation.
Now, in $0 < x \leq \pi,\ x^{3} + (x + 2)^{2} + 2\sin{x > 4}$
and in $\pi < x \leq 2\pi,\ x^{3} + (x + 2)^{2} + 2\sin{x > 27 + 25 - 2 = 50}$
Hence, $x = 0$ is the only solution