If $x$ is real,then the maximum and minimum values of the expression $\frac{x^2 + 14x + 9}{x^2 + 2x + 3}$ are

If $y = \frac{x^2 + 14x + 9}{x^2 + 2x + 3}$ for all $x \in R$,then the interval of maximum length in which $y$ lies is

Let $[r]$ denote the greatest integer not exceeding $r$. The roots of the equation $3 x^2 + 6 x + 5 + \alpha (x^2 + 2 x + 2) = 0$ are complex numbers whenever $\alpha > L$ or $\alpha < M$. If $(L - M)$ is minimum,then the greatest value of $[r]$ such that $L y^2 + M y + r < 0$ for all $y \in R$ is:

If $\left|\frac{x^2+kx+1}{x^2+x+1}\right| < 3$ for all real $x$,then $k$ is in the interval

If $a, b, c$ are real and $x^3 - 3b^2x + 2c^3$ is divisible by $x - a$ and $x - b$,then:

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