The inequality provided in the image is:

$x^2 + 2x + 1 < 0$

Step-by-step Solution:

1. Factor the quadratic expression:
   $x^2 + 2x + 1 = (x + 1)^2$ This simplifies to: $(x + 1)^2 < 0$

2. Analyze the inequality:
   The square of any real number is always non-negative, i.e., $(x + 1)^2 \geq 0$ for all real values of $x$. Therefore, there is no real solution to this inequality because a square can never be strictly less than zero.

Conclusion:

The inequality $(x + 1)^2 < 0$ has no real solutions. This means that there is no value of $x$ for which $x^2 + 2x + 1 < 0$.

Would you like more details or have any questions?

Related Questions:

1. How do you solve inequalities involving quadratic expressions with real solutions?
2. What does the graph of $(x + 1)^2 = 0$ look like, and how does it help in understanding inequalities?
3. Can quadratic inequalities have complex solutions, and how are they interpreted?
4. How would you approach solving a quadratic inequality when it has distinct real roots?
5. What is the difference between strict inequalities (e.g., $<$) and non-strict inequalities (e.g., $\leq$)?

Tip:

When dealing with quadratic inequalities, always factor and check the nature of the solutions (real or complex) before jumping to conclusions. This can save time when interpreting the solution set.