To solve the equation $x^2 + 6x - 3 = 0$ by completing the square, let's go through the steps:

1. Start with the original equation: $x^2 + 6x - 3 = 0$

2. Move the constant term to the other side: $x^2 + 6x = 3$

3. To complete the square, take half of the coefficient of $x$, which is 6, divide by 2 to get 3, and then square it to get $9$.

4. Add $9$ to both sides of the equation to complete the square: $x^2 + 6x + 9 = 3 + 9$ $x^2 + 6x + 9 = 12$

5. Now, rewrite the left side as a perfect square: $(x + 3)^2 = 12$

6. Take the square root of both sides, remembering to consider both the positive and negative roots: $x + 3 = \pm \sqrt{12}$ Simplify $\sqrt{12}$ as $2\sqrt{3}$: $x + 3 = \pm 2\sqrt{3}$

7. Solve for $x$ by isolating it: $x = -3 \pm 2\sqrt{3}$

So, the solutions are: $x = -3 + 2\sqrt{3} \quad \text{and} \quad x = -3 - 2\sqrt{3}$

Would you like further details or have any questions about the steps?

Here are five related questions for further exploration:

1. How does completing the square help in solving quadratic equations?
2. What are other methods to solve quadratic equations besides completing the square?
3. How can we verify the solutions of a quadratic equation?
4. What is the significance of the discriminant in a quadratic equation?
5. How would the process change if the equation had a different leading coefficient?

Tip: Completing the square is also a useful technique for converting quadratic equations into vertex form, which reveals the maximum or minimum point of the parabola.