The problem asks to solve the quadratic equation $x^2 + 10x + 9 = 0$ by completing the square. Let's walk through the steps:

Step 1: Rewrite the equation

We start with: $x^2 + 10x + 9 = 0$

Move the constant term (9) to the other side: $x^2 + 10x = -9$

Step 2: Complete the square

To complete the square, take half of the coefficient of $x$ (which is 10), square it, and add it to both sides. Half of 10 is 5, and $5^2 = 25$.

Add 25 to both sides: $x^2 + 10x + 25 = -9 + 25$ $(x + 5)^2 = 16$

Step 3: Solve for $x$

Now take the square root of both sides: $x + 5 = \pm 4$

Step 4: Solve for $x$

Subtract 5 from both sides to find the solutions: $x = -5 + 4 = -1 \quad \text{or} \quad x = -5 - 4 = -9$

Thus, the solutions are: $x = -1 \quad \text{or} \quad x = -9$

The correct answer is the last option: $x = -1$ or $x = -9$.

Would you like more details on any part of this solution?

Related questions:

1. How do you derive the quadratic formula from completing the square?
2. Can all quadratic equations be solved by completing the square?
3. What is the difference between completing the square and factoring?
4. How can you verify the solutions to a quadratic equation?
5. Why is completing the square useful in solving quadratic equations?

Tip: Always check if a quadratic equation can be factored easily before completing the square, as factoring might be quicker!