Explanation

(a)
Let $y = \frac{x^{2} - x + 1}{x^{2} + x + 1}$
$\Rightarrow \ \ \ x^{2}(y - 1) + x(y + 1) + 1(y - 1) = 0$
Here, $D \geq 0$ as $x$ is real
$\therefore\ \ \ (y + 1)^{2} - 4(y - 1)^{2} \geq 0$
$\Rightarrow \ \ \ y^{2} + 2y + \left( 1 - 4y^{2} + 1 - 2y \right) \geq 0$
$\Rightarrow \ \ \ - 3y^{2} - 10y + 3 \geq 0$
$\Rightarrow 3y^{2} - 10y + 3 \leq 0$
$\Rightarrow (3y - 1)(y - 3) \leq 0$
$\Rightarrow \ \frac{1}{3} \leq y \leq 3$