The number of distinct real roots of the equation $|\mathrm{x}||\mathrm{x}+2|-5|\mathrm{x}+1|-1=0$ is....................

Below are four equations in $x$. Assume that $0 < r < 4$. Which of the following equations has the largest solution for $x$ ?

Let, $\alpha, \beta$ be the distinct roots of the equation $\mathrm{x}^2-\left(\mathrm{t}^2-5 \mathrm{t}+6\right) \mathrm{x}+1=0, \mathrm{t} \in \mathrm{R}$ and $\mathrm{a}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}$. Then the minimum value of $\frac{\mathrm{a}_{2023}+\mathrm{a}_{2025}}{\mathrm{a}_{2024}}$ is

The number of distinct real roots of $x^4-4 x^3+12 x^2+x-1=0$ is

The number of roots of the equation $\log ( - 2x)$ $ = 2\log (x + 1)$ are

The equation $e^{4 x}+8 e^{3 x}+13 e^{2 x}-8 e^x+1=0, x \in R$ has: