1. Rewrite the function

Given

Factor the common terms:

So,

2. Check which factors can be zero

• Since $e^{\pi x}>0$ for all $x\in\mathbb R$, it is never zero.
• Consider $x^2-x+3=\left(x-\frac12\right)^2+\frac{11}{4}>0$ for all real $x$, so this is never zero.
• Also, $-1\le \sin x \le 1 \implies 1\le \sin x+2 \le 3,$ hence $\sin x+2>0$ for all real $x$.

Therefore, the only way $f(x)=0$ is when

3. Find the number of real roots of $x^{2023}+2024x+2025=0$

Let

Then

Since $x^{2022}\ge 0$ for all real $x$,

So $g(x)$ is strictly increasing on $\mathbb R$.

Now check end behavior:

A strictly increasing continuous function going from $-\infty$ to $+\infty$ crosses the x-axis exactly once.

Hence, the equation

has exactly one real solution.

4. Conclusion

Therefore, the number of real solutions of $f(x)=0$ is $\boxed{1}.$