Solve the given two equations and select the correct answer from the given options.
$I. x^{2} + 12x + 32 = 0$
$II. 2y^{2} + 15y + 27 = 0$

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + qx - r = 0$,then find the equation whose roots are $\left( \beta \gamma + \frac{1}{\alpha} \right), \left( \gamma \alpha + \frac{1}{\beta} \right), \left( \alpha \beta + \frac{1}{\gamma} \right)$.

If the roots of the equation $x^2 - 5x + 16 = 0$ are $\alpha, \beta$ and the roots of the equation $x^2 + px + q = 0$ are $\alpha^2 + \beta^2$ and $\frac{\alpha \beta}{2}$,then:

All the pairs $(x, y)$ that satisfy the inequality $2^{\sqrt{\sin^2 x - 2\sin x + 5}} \cdot \frac{1}{4^{\sin^2 y}} \leq 1$ also satisfy the equation:

If $\alpha, \beta, \gamma$ are roots of $x^3 - 2x^2 + 3x - 2 = 0$,then the value of $\left( \frac{\alpha\beta}{\alpha + \beta} + \frac{\alpha\gamma}{\alpha + \gamma} + \frac{\beta\gamma}{\beta + \gamma} \right)$ is

The sum of the roots of a quadratic equation is $2$ and the sum of their cubes is $98$. Then the equation is: