1. **State the problem:** Solve the inequality $x^2 + 1 < 0$. 2. **Recall the properties of squares:** For any real number $x$, $x^2 \geq 0$. This means $x^2$ is always non-negative. 3. **Analyze the inequality:** Since $x^2 \geq 0$, the smallest value $x^2 + 1$ can take is $1$ (when $x=0$). 4. **Check if the inequality can be true:** The inequality $x^2 + 1 < 0$ asks for values of $x$ such that $x^2 + 1$ is less than zero. 5. **Conclusion:** Since $x^2 + 1 \geq 1$ for all real $x$, it is never less than zero. Therefore, there are no real solutions to the inequality. **Final answer:** No real values of $x$ satisfy $x^2 + 1 < 0$.