To solve the equation $5(x + 11)^2 = -50$, we can follow these steps:

1. Divide both sides by 5: $(x + 11)^2 = -10$

2. Analyze the equation: The left side, $(x + 11)^2$, represents a square and cannot be negative. Therefore, there are no real solutions to this equation.

3. Complex solutions: If we allow complex numbers, we can proceed by taking the square root of both sides: $x + 11 = \pm \sqrt{-10}$ This can be expressed as: $x + 11 = \pm i\sqrt{10}$

4. Isolate $x$: $x = -11 \pm i\sqrt{10}$

Thus, the solutions are: $x = -11 + i\sqrt{10} \quad \text{and} \quad x = -11 - i\sqrt{10}$

If you have any questions or need further details, feel free to ask! Here are some related questions to consider:

1. What does it mean for a solution to be complex?
2. How can we graph complex solutions on the complex plane?
3. What are some real-world applications of complex numbers?
4. How do you solve other quadratic equations that might yield complex solutions?
5. What are the properties of complex numbers that are relevant in mathematics?

Tip: Always check if the equation can yield real solutions before proceeding with complex solutions.