Solve the given two equations and select the correct answer from the given options.
$I.$ $8x^2 + 6x = 5$
$II.$ $12y^2 - 22y + 8 = 0$

Let $\alpha$ and $\beta$ be the roots of $5x^2 - 3x - 1 = 0$. Then,the expression $\left[ (\alpha + \beta)x - \left( \frac{\alpha^2 + \beta^2}{2} \right)x^2 + \left( \frac{\alpha^3 + \beta^3}{3} \right)x^3 - \dots \right]$ is equal to:

The number of solutions of $\log_4(x - 1) = \log_2(x - 3)$ is:

Solve the given two equations and select the correct option.
$I.$ $77 x^{2}+58 x+8=0$
$II.$ $42 y^{2}+59 y+20=0$

When $a = -5, b = -6$ and $c = 10$,find the value of $\frac{a^{3} + b^{3} + c^{3} - 3abc}{ab + bc + ca - a^{2} - b^{2} - c^{2}}$.

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 2(m^2 + 1)x + m^4 + m^2 + 1 = 0$,then $\alpha^2 + \beta^2$ equals: