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 $b= \lim _{\mathrm{x} \rightarrow 0}\left(\frac{\int_0^{\mathrm{x}} \frac{\log (1+\mathrm{t})}{\mathrm{t}^{2024+1}} \mathrm{dt}}{\mathrm{x}^2}\right)$. If the equations $c x^2+d x+e=0$ and $2 b x^2+a x+4=0$ have a common root, where $c, d, \mathrm{e} \in \mathrm{R}$, then $\mathrm{d}: \mathrm{c}: \mathrm{e}$ equals