Let a be the sum of all coefficients in the expansion of ( 1 – 2x +...

Given: $a$ is the sum of all coefficients in ${\left(1 - 2x + 2{x}^{2}\right)}^{2023}{\left(3 - 4{x}^{2} + 2{x}^{3}\right)}^{2024}$

$⇒ a = {\left(1 - 2 \times 1 + 2 \times 1\right)}^{2023}{\left(3 - 4 \times 1 + 2 \times 1\right)}^{2024}$

$⇒ a = 1 . . . \left(i\right)$

Now, $b = \lim_{x \to 0}\dfrac{{∫}_{0}^{x}\dfrac{\log \left(1 + t\right)}{{t}^{2024} + 1}dt}{{x}^{2}}$

Using L-Hospital's rule and Newton Leibnitz Theorem, we get

$⇒ b = \lim_{x \to 0}\dfrac{\log \left(1 + x\right)}{\left({x}^{2024} + 1\right)2x}$

$⇒ b = \lim_{x \to 0}\dfrac{1}{2\left({x}^{2024} + 1\right)}$

$⇒ b = \dfrac{1}{2} . . . \left(ii\right)$

Also, $2b{x}^{2} + ax + 4 = 0$

$⇒ {x}^{2} + x + 4 = 0$, which gives complex conjugates as roots.

Let $\alpha\text{and} \alpha$ be those roots.

Then, $c{x}^{2} + dx + e = 0$ will also have $\alpha\text{and} \alpha$ as roots.

$⇒ d: c: e = 1: 1: 4$