For any real value of $x$,if $\frac{11 x^2+12 x+6}{x^2+4 x+2} \notin(a, b]$,then the value of $x$ for which $\frac{11 x^2+12 x+6}{x^2+4 x+2}=b-a+3$ is

Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^2+20x-2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^2-20x+2020$. Then the value of $ac(a-c)+ad(a-d)+bc(b-c)+bd(b-d)$ is

The number of ordered pairs $(a, b)$ of integers such that $a-b$ is a root of $x^2+ax+b=0$ is

The equation of lowest degree with rational coefficients having roots $\sqrt{3}+\sqrt{2} i$ and $\sqrt{3}-\sqrt{2}$ is

For real $x$, the greatest value of $\frac{x^{2}+2 x+4}{2 x^{2}+4 x+9}$ is

Let $\alpha, \beta$ be the roots of the equation $x^2 - x + p = 0$ and $\gamma, \delta$ be the roots of the equation $x^2 - 4x + q = 0$, where $p, q \in Z$. If $\alpha, \beta, \gamma, \delta$ are in $G$.$P$.,then $|p + q|$ equals: