For real values of $x$,the range of $\frac{x^2+2x+1}{x^2+2x-1}$ is

Suppose $a, b, c$ are three distinct real numbers. Let $P(x) = \frac{(x-b)(x-c)}{(a-b)(a-c)} + \frac{(x-c)(x-a)}{(b-c)(b-a)} + \frac{(x-a)(x-b)}{(c-a)(c-b)}$. When simplified,$P(x)$ becomes:

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

Let $x$ be a real number. Match the following:

List-$I$	List-$II$
$(A)$ The minimum value of $2x^2 + 4x + 5$	$(I)$ $-1$
$(B)$ The maximum value of $\frac{x^2 + 4x + 1}{x^2 + x + 1}$	$(II)$ $1$
$(C)$ If $1 \leq \frac{3x^2 - 5x + 6}{x^2 + 1} \leq 2$,$\forall x \in [a, b]$ then $b =$	$(III)$ $2$
$(D)$ If $1 \leq \frac{3x^2 - 5x + 6}{x^2 + 1} \leq 2$,$\forall x \in [a, b]$ then $a =$	$(IV)$ $3$
	$(V)$ $4$

If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c = 0$,and $\alpha + \beta$,$\alpha^2 + \beta^2$,and $\alpha^3 + \beta^3$ are in a geometric progression,and $\Delta = b^2 - 4ac$,then which of the following is true?

The number of solutions of the equations $x+y+z=12$,$x^2+y^2+z^2=50$,and $x^3+y^3+z^3=216$ is