If $z_1, z_2, z_3, z_4$ are the roots of the equation $z^4 + z^3 + z^2 + z + 1 = 0$,then $\prod_{i=1}^{4} (z_i + 2)$ is equal to:

$I. x^{2}+10 x+24=0$
$II. 4 y^{2}-17 y+18=0$

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{4} - 227 = 398$
$II.$ $y^{2} + 321 = 346$

Let $y = f(x) = ax^2 + 2bx + c$ (where $a, b, c \in R$ and $a \neq 0$). If $f(x) = 0$ has imaginary roots and $4a + 4b + c < 0$,then which of the following is true?

$P$ can complete a piece of work in $16$ days,which $Q$ can complete in $32$ days. If $P$ and $Q$ work on alternate days,find the relationship between Quantity $I$ (Total days taken if $P$ starts) and Quantity $II$ (Total days taken if $Q$ starts).

Solve the given two equations and provide the correct answer from the given options.
$I.$ $391 x^{2} + 1344 x + 1073 = 0$
$II.$ $437 y^{2} + 1074 y + 589 = 0$