Find a polynomial $f(x)$ of degree $2$ where $f(0)=8, f(1)=12, f(2)=18$.

The number of real solutions of the equation $|x^2 + 4x + 3| + 2x + 5 = 0$ is:

Suppose $m, n$ are positive integers such that $6^m + 2^{m+n} \cdot 3^w + 2^n = 332$. The value of the expression $m^2 + mn + n^2$ is

The number of real roots of the equation $e^{4x} - e^{3x} - 4e^{2x} - e^{x} + 1 = 0$ is equal to $.....$

The roots of the equation $x^3-3x-2=0$ 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 $S$?
$(A) \left(-\frac{1}{2}, -\frac{1}{\sqrt{5}}\right)$
$(B) \left(-\frac{1}{\sqrt{5}}, 0\right)$
$(C) \left(0, \frac{1}{\sqrt{5}}\right)$
$(D) \left(\frac{1}{\sqrt{5}}, \frac{1}{2}\right)$