If $x+y+z=0,$ then find the value of $\frac{(x+y)(y+z)(z+x)}{x y z}$

Solve the given two equations and select the correct answer from the given options.
$I.$ $4x + 7y = 209$
$II.$ $12x - 14y = -38$

If $\alpha$ and $\beta$ are roots of the equation $x^2 - 4\sqrt{2}kx + 2e^{4\ln k} - 1 = 0$ for some $k$, and $\alpha^2 + \beta^2 = 66$, then $\alpha^3 + \beta^3$ is equal to: (in $\sqrt{2}$)

If the difference of the roots of $x^2 - px + 8 = 0$ is $2$,then the value of $p$ is

$\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}$ is equal to

The condition for the roots of the equation $(c^2 - ab)x^2 - 2(a^2 - bc)x + (b^2 - ac) = 0$ to be equal is