If $ax^2 + bx + c = 0$ has real and distinct roots,$\alpha$ and $\beta$ where $\beta > \alpha$. Further,if $a > 0, b < 0$,and $c < 0$,then:

Let $f(x)$ be a polynomial and $a, b$ be distinct real numbers. Then the remainder in the division of $f(x)$ by $(x-a)(x-b)$ is

If roots of the equation $ax^2 + bx + c = 0$ are $(\alpha - \beta)$ and $(\gamma - \delta)$,and roots of the equation $Ax^2 + Bx + C = 0$ are $(\alpha + \delta)$ and $(\beta + \gamma)$,then $\left| \frac{a}{A} \right|$ is equal to (where $D_1$ and $D_2$ are discriminants of the given equations respectively).

The number of distinct real roots of the equation $x^{5}(x^{3}-x^{2}-x+1)+x(3x^{3}-4x^{2}-2x+4)-1=0$ is

Consider the equation $(1+a+b)^2=3(1+a^2+b^2)$ where $a, b$ are real numbers. Then,

Assertion $(A)$: The maximum value of $-x^2+3x+1$ is $\frac{13}{4}$.
Reason $(R)$: If $a < 0$,the maximum value of $ax^2+bx+c$ exists at $x = -\frac{b}{2a}$.
The correct option among the following is