To solve the equation $2x^2 + 12x + 3 = 15$ by completing the square, follow these steps:

1. Move the constant to the right-hand side: $2x^2 + 12x + 3 - 3 = 15 - 3$ This simplifies to: $2x^2 + 12x = 12$

2. Factor out the coefficient of $x^2$ (which is 2): $2(x^2 + 6x) = 12$

3. Divide both sides by 2 to simplify: $x^2 + 6x = 6$

4. Complete the square: Take half of the coefficient of $x$ (which is 6), divide it by 2, and then square it. $\left(\frac{6}{2}\right)^2 = 9$ Add 9 to both sides of the equation: $x^2 + 6x + 9 = 6 + 9$ Simplifies to: $(x + 3)^2 = 15$

5. Solve for $x$ by taking the square root of both sides: $x + 3 = \pm \sqrt{15}$ Therefore: $x = -3 \pm \sqrt{15}$

Final answer:

$x = -3 \pm \sqrt{15}$

The missing pieces in your format:

• $\boxed{-3}$
• $\boxed{15}$

Would you like more details on any of the steps? Here are 5 related questions you might find useful:

1. How does completing the square differ from using the quadratic formula?
2. Why do we add the square of half the linear coefficient when completing the square?
3. Can you always use completing the square for any quadratic equation?
4. How would the solution change if the equation was set to 0 instead of 15?
5. How do you handle completing the square when the quadratic coefficient isn't 1?

Tip: Completing the square is particularly useful when you want to convert a quadratic equation into vertex form!