Determine $p$ such that the equation $x^{2}+5px+16=0$ has no real roots.

If $\alpha$ and $\beta$ are the roots of the equation ${x^2} - a(x + 1) - b = 0$,then $(\alpha + 1)(\beta + 1) = $

Suppose that $x$ and $y$ are positive numbers with $xy = \frac{1}{9}$,$x(y + 1) = \frac{7}{9}$,and $y(x + 1) = \frac{5}{18}$. The value of $(x + 1)(y + 1)$ equals:

If $\alpha, \beta$ and $\gamma$ are three consecutive terms of a non-constant $G.P.$ such that the equations $\alpha x^2 + 2\beta x + \gamma = 0$ and $x^2 + x - 1 = 0$ have a common root,then $\alpha(\beta + \gamma)$ is equal to

Solve the given two equations and provide the correct answer from the given options.
$I.$ $x^{2}+14x+49=0$
$II.$ $y^{2}+9y=0$

Solve the given two equations and select the correct answer from the given options.
$I.$ $7x = 4y + 85$
$II.$ $y = \sqrt[3]{17576}$