The number of distinct real solutions of the equation $x|x+4|+3|x+2|+10=0$ is

Let $f(x)$ be a quadratic polynomial with $f(2)=10$ and $f(-2)=-2$. Then,the coefficient of $x$ in $f(x)$ is

Let $S = \{x^{3} + ax^{2} + bx + c : a, b, c \in \mathbb{N} \text{ and } a, b, c \le 20\}$ be a set of polynomials. Then the number of polynomials in $S$, which are divisible by $x^{2} + 2$, is

Both the roots of the given equation $(x - a)(x - b) + (x - b)(x - c) + (x - c)(x - a) = 0$ are always

In the equation $x^3 + 3Hx + G = 0$,if $G$ and $H$ are real and $G^2 + 4H^3 > 0$,then the roots are

Let $S$ be the set of all non-zero real numbers $\alpha$ such that the quadratic equation $\alpha x^2 - x + \alpha = 0$ has two distinct real roots $x_1$ and $x_2$ satisfying the inequality $|x_1 - x_2| < 1$. Which of the following intervals is(are) a subset$(s)$ of $S$?
$(A) \left(-\frac{1}{2}, -\frac{1}{\sqrt{5}}\right)$
$(B) \left(-\frac{1}{\sqrt{5}}, 0\right)$
$(C) \left(0, \frac{1}{\sqrt{5}}\right)$
$(D) \left(\frac{1}{\sqrt{5}}, \frac{1}{2}\right)$