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The question in **Problem 26** asks:

"What number must be added to \( x^2 + 4x + 1 \) to form a perfect square?"

To form a perfect square trinomial, we can use the process of completing the square. Here's the step-by-step method:

### Step 1: Identify the coefficient of the linear term.
The given quadratic is \( x^2 + 4x + 1 \). The coefficient of the linear term \( 4x \) is \( 4 \).

### Step 2: Divide the linear coefficient by 2 and square it.
Take half of the linear coefficient \( 4 \), which is:
\[
\frac{4}{2} = 2
\]
Now, square it:
\[
2^2 = 4
\]

### Step 3: Adjust the constant term.
To complete the square, the constant term should be \( 4 \), but the current constant term is \( 1 \).

Thus, the number that must be **added** is:
\[
4 - 1 = 3
\]

### Final Answer: 
The number that must be added is **3** (Option D).

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Would you like more details or explanations?

Here are 5 related questions to explore:
1. What is the process of completing the square for different types of quadratics?
2. How do you identify perfect square trinomials quickly?
3. What happens if the coefficient of \( x^2 \) is not 1 when completing the square?
4. Can completing the square be used to solve quadratic equations?
5. How does completing the square relate to the quadratic formula?

**Tip:** Completing the square is especially useful for converting quadratic equations into vertex form, which makes graphing easier!

What number must be added to x^2 + 4x + 1 to form a perfect square?

Learn how to find the number needed to add to x^2 + 4x + 1 to complete the square. This problem demonstrates key algebraic techniques, including completing the square. Suitable for students in Grades 9-11.

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