If ${x^2} + 2x + 2xy + my - 3$ has two rational factors,then the value of $m$ will be

Let $a \neq 0$ and $p(x)$ be a polynomial of degree greater than $2$. If $p(x)$ leaves remainders $a$ and $-a$ when divided respectively by $x+a$ and $x-a$,then the remainder when $p(x)$ is divided by $x^2-a^2$ is:

Let $p, q, r \in \mathbb{R}$ and $r > p > 0$. If the quadratic equation $px^2 + qx + r = 0$ has two complex roots $\alpha$ and $\beta$,then what is the value of $|\alpha| + |\beta|$?

If $k \in (-\infty, -2) \cup (2, \infty)$,then the roots of the equation $x^2 + 2kx + 4 = 0$ are

Let $\alpha, \beta$ be two roots of the quadratic equation $x^2+ax-b=0, b \neq 0$. If the straight line $x \cos \theta + y \sin \theta = c$ touches the curve $(\frac{x}{\alpha})^n + (\frac{y}{\beta})^n = 2$ at the point $(\alpha, \beta)$, then $(\frac{a}{b})^2 + \frac{2}{b} =$

Let the roots of the equation $E_1 \equiv x^3+x^2+lx+n=0$ be $x_i, (i=1, 2, 3)$ and the roots of $E_2 \equiv x^3+ax^2+bx+c=0$ be $\frac{x_i-1}{2}$. If the equation $E_2=0$ is a reciprocal equation of class one,then the roots of these two equations excluding the common roots are