Let $\alpha$ and $\beta$ be the roots of the equation $(x - a)(x - b) = c$,where $c \neq 0$. What are the roots of the equation $(x - \alpha)(x - \beta) + c = 0$?

If $x^2 + px + 1$ is a factor of $ax^3 + bx + c$,then

Let $p(x) = x^2 + ax + b$ have two distinct real roots,where $a, b$ are real numbers. Define $g(x) = p(x^3)$ for all real numbers $x$. Then,which of the following statements are true?
$I.$ $g$ has exactly two distinct real roots.
$II.$ $g$ can have more than two distinct real roots.
$III.$ There exists a real number $\alpha$ such that $g(x) \geq \alpha$ for all real $x$.

For what values of $a \in \mathbb{Z}$,the quadratic expression $(x+a)(x+1991)+1$ can be factorised as $(x+b)(x+c)$,where $b, c \in \mathbb{Z}$?

If $3$ is a root of $x^2+kx-24=0$,then it is also a root of which of the following equations?

The number of solutions of the equation $e^{\sin x} - 2e^{-\sin x} = 2$ is