The set of all real values $a$ for which $-1 < \frac{2 x^2+a x+2}{x^2+x+1} < 3$ holds for all real values of $x$ is

Let $f(x)=a x^2+b x+c$ and the $GCD$ of $a, b, c$ be $1$. If $\frac{-7+\sqrt{11} i}{6}$ is a root of $f(x)=0$ and $f\left(\frac{x}{k}\right)-L=(x+4)(3 x-5)$,then $k$ and $L$ are respectively:

What is the minimum value of $\frac{1 - x + x^2}{1 + x + x^2}$?

For what condition will the expression $a^2x^2 + bx + 1$ be positive for all $x \in R$?

Let $\phi(x)=\frac{x}{(x^2+1)(x+1)}$. If $a, b$ and $c$ are the roots of the equation $x^3-3x+\lambda=0, (\lambda \neq 0)$,then $\phi(a) \phi(b) \phi(c) =$

The number of ordered pairs $(x, y)$ of positive integers satisfying $2^x + 3^y = 5^{xy}$ is