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We are asked to express \( 2x^2 + 12x + 6 \) in the form \( a(x + b)^2 + c \), and find the values of \( a \), \( b \), and \( c \).

Let's complete the square step by step.

### Step 1: Factor out the coefficient of \( x^2 \) from the first two terms:
\[
2x^2 + 12x + 6 = 2(x^2 + 6x) + 6
\]

### Step 2: Complete the square inside the parentheses:
Take half of the coefficient of \( x \) (which is 6), square it, and add/subtract it inside the parentheses.
\[
x^2 + 6x \quad \text{(Half of 6 is 3, and \( 3^2 = 9 \))}
\]
Add and subtract 9 inside the parentheses:
\[
2(x^2 + 6x + 9 - 9) + 6 = 2((x + 3)^2 - 9) + 6
\]

### Step 3: Simplify the expression:
Distribute the 2:
\[
2((x + 3)^2 - 9) + 6 = 2(x + 3)^2 - 18 + 6 = 2(x + 3)^2 - 12
\]

Thus, the expression is now in the desired form \( a(x + b)^2 + c \), where:
\[
a = 2, \quad b = 3, \quad c = -12
\]

### Final Answer:
- \( a = 2 \)
- \( b = 3 \)
- \( c = -12 \)

Would you like further details or have any questions? Here are 5 questions that expand on this concept:

1. How do you complete the square for more complicated quadratic expressions?
2. What are some applications of completing the square in solving quadratic equations?
3. How does completing the square relate to the vertex form of a quadratic equation?
4. Can completing the square be used for quadratic inequalities? If so, how?
5. What is the geometric interpretation of the vertex form \( a(x + b)^2 + c \)?

**Tip:** Completing the square is especially useful for converting quadratic equations into vertex form, making it easier to identify key features like the vertex of a parabola.

Express 2x^2 + 12x + 6 in the form a(x + b)^2 + c, where a, b, and c are numbers. What are the values of a, b, and c?

Learn how to express the quadratic expression 2x^2 + 12x + 6 in the form a(x + b)^2 + c using the completing the square method. This problem covers key algebraic concepts such as factoring and completing the square, and is suitable for students in Grades 9-11.

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