Visual guide to completing the square using a geometric model, showing how x² + 4x + 4 equals (x + 2)².

Let's solve the problem step by step, based on the image you've provided.

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Understanding the Task

The task is to complete the square for several quadratic expressions. Completing the square means rewriting a quadratic expression in the form:

$$
(x + a)^2 = x^2 + 2ax + a^2
$$

We are given three expressions to complete the square:

1) $ x^2 + 6x + 9 $
2) $ x^2 + 8x + 16 $
3) $ x^2 + 10x + 25 $

And then we are asked:
What would the next expression in this sequence be? Explain.

Also, there's a challenge:
Can you turn $ 4x^2 + 12x + 9 $ into a square?

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Step 1: Complete the Square for Each Expression

Let’s examine each one.

#### 1) $ x^2 + 6x + 9 $

We look at the middle term: $ 6x $. Half of 6 is 3, and $ 3^2 = 9 $. So:

$$
x^2 + 6x + 9 = (x + 3)^2
$$

✔ This is a perfect square.

#### 2) $ x^2 + 8x + 16 $

Half of 8 is 4, $ 4^2 = 16 $:

$$
x^2 + 8x + 16 = (x + 4)^2
$$

✔ Perfect square.

#### 3) $ x^2 + 10x + 25 $

Half of 10 is 5, $ 5^2 = 25 $:

$$
x^2 + 10x + 25 = (x + 5)^2
$$

✔ Perfect square.

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Step 2: Find the Next Expression in the Sequence

Now let’s look at the pattern:

- $ x^2 + 6x + 9 = (x+3)^2 $
- $ x^2 + 8x + 16 = (x+4)^2 $
- $ x^2 + 10x + 25 = (x+5)^2 $

So the pattern is:

- The coefficient of $ x $ increases by 2 each time: 6 → 8 → 10 → ?
- The constant term is the square of half the coefficient of $ x $:
- For $ 6x $: $ (6/2)^2 = 3^2 = 9 $
- For $ 8x $: $ (8/2)^2 = 4^2 = 16 $
- For $ 10x $: $ (10/2)^2 = 5^2 = 25 $

So the next one should have:

- Coefficient of $ x $: $ 12 $
- Constant term: $ (12/2)^2 = 6^2 = 36 $

Thus, the next expression is:

$$
x^2 + 12x + 36 = (x + 6)^2
$$

✔ Answer: The next expression in the sequence is $ x^2 + 12x + 36 $, because it follows the pattern where the linear coefficient increases by 2 each time, and the constant term is the square of half that coefficient.

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Step 3: Challenge – Can you turn $ 4x^2 + 12x + 9 $ into a square?

This is not a standard monic quadratic (leading coefficient ≠ 1), so we need to handle it carefully.

Let’s try to write it as a square.

We suspect it might be a perfect square trinomial.

Try factoring or completing the square.

Let’s factor:

$$
4x^2 + 12x + 9
$$

We can try to write this as $ (ax + b)^2 $

Assume:
$$
(ax + b)^2 = a^2x^2 + 2abx + b^2
$$

Compare with $ 4x^2 + 12x + 9 $:

- $ a^2 = 4 $ → $ a = 2 $
- $ b^2 = 9 $ → $ b = 3 $
- Check middle term: $ 2ab = 2 \cdot 2 \cdot 3 = 12 $ ✔

Perfect!

So:
$$
4x^2 + 12x + 9 = (2x + 3)^2
$$

✔ Yes! It can be turned into a square.

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Visual Aid from Image

The diagram shows:

- A large square of side $ x + 2 $
- Divided into:
- $ x^2 $ (green)
- Two rectangles of $ x \times 2 $ (yellow)
- One $ 2 \times 2 $ square (red)

So total area:
$$
x^2 + 2x + 2x + 4 = x^2 + 4x + 4 = (x + 2)^2
$$

This visually demonstrates completing the square.

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✔ Final Answers

#### Complete the squares:
1) $ x^2 + 6x + 9 = (x + 3)^2 $
2) $ x^2 + 8x + 16 = (x + 4)^2 $
3) $ x^2 + 10x + 25 = (x + 5)^2 $

#### Next expression in the sequence:
$$
x^2 + 12x + 36 = (x + 6)^2
$$
Because the pattern is increasing the linear coefficient by 2 and squaring half of it.

#### Challenge:
Yes,
$$
4x^2 + 12x + 9 = (2x + 3)^2
$$

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Let me know if you'd like a visual explanation or further practice!