For $x \in \mathbb{R}$,the minimum value of $\frac{x^2+2x+5}{x^2+4x+10}$ is

Let $\alpha, \beta (\alpha > \beta)$ be the roots of the quadratic equation $x^{2} - x - 4 = 0$. If $P_{n} = \alpha^{n} - \beta^{n}, n \in N$,then $\frac{P_{15} P_{16} - P_{14} P_{16} - P_{15}^{2} + P_{14} P_{15}}{P_{13} P_{14}}$ is equal to $......$

$\sqrt{x + 2\sqrt{x - 1}} + \sqrt{x - 2\sqrt{x - 1}} = $

Let $\alpha, \beta$ be the roots of the equation $x^2 - 3x + r = 0$, and $\frac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2 + 3x + r = 0$. If the roots of the equation $x^2 + 6x = m$ are $2\alpha + \beta + 2r$ and $\alpha - 2\beta - \frac{r}{2}$, then $m$ is equal to:

If a polynomial $P(x)$ of degree $4$ is given by $P(x) = 2x^4 + ax^3 + bx^2 + cx + d$ such that $P(1) = 4, P(2) = 7, P(3) = 12$,and $P(4) = 19$,then find the value of $P(5)$.

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