Let $a$ be the sum of all coefficients in the expansion of $\left( 1 - 2 x + 2 x ^ { 2 } \right) ^ { 2023 } \left( 3 - 4 x ^ { 2 } + 2 x ^ { 3 } \right) ^ { 2024 }$ and $b = \lim _ { x \rightarrow 0 } \frac { \int _ { 0 } ^ { x } \frac { \log ( 1 + t ) } { t ^ { 2024 } + 1 } dt } { x ^ { 2 } }$. If the equations $c x ^ { 2 } + d x + e = 0$ and $2 b x ^ { 2 } + a x + 4 = 0$ have a common root, where $c , d , e \in R$, then $d : c : e$ equals (1) $2 : 1 : 4$ (2) $4 : 1 : 4$ (3) $1 : 2 : 4$ (4) $1 : 1 : 4$
Let $a$ be the sum of all coefficients in the expansion of $\left( 1 - 2 x + 2 x ^ { 2 } \right) ^ { 2023 } \left( 3 - 4 x ^ { 2 } + 2 x ^ { 3 } \right) ^ { 2024 }$ and $b = \lim _ { x \rightarrow 0 } \frac { \int _ { 0 } ^ { x } \frac { \log ( 1 + t ) } { t ^ { 2024 } + 1 } dt } { x ^ { 2 } }$. If the equations $c x ^ { 2 } + d x + e = 0$ and $2 b x ^ { 2 } + a x + 4 = 0$ have a common root, where $c , d , e \in R$, then $d : c : e$ equals\\
(1) $2 : 1 : 4$\\
(2) $4 : 1 : 4$\\
(3) $1 : 2 : 4$\\
(4) $1 : 1 : 4$