1. **State the problem:** Solve the inequality $x^2 + 1 > 0$. 2. **Recall the properties:** The expression $x^2$ is always non-negative for all real $x$, since squaring any real number results in a value $\geq 0$. 3. **Analyze the inequality:** Since $x^2 \geq 0$, then $x^2 + 1 \geq 1$ for all real $x$. 4. **Conclusion:** Because $x^2 + 1$ is always at least 1, it is always greater than 0 for every real number $x$. 5. **Final answer:** The inequality $x^2 + 1 > 0$ holds for all real numbers $x$. Therefore, the solution set is $\boxed{(-\infty, \infty)}$.