https://file.aiquickdraw.com/mathbot/file/1726748759914stvo76tl.jpg

This puzzle appears to involve filling in the red "Empty Box" spaces with mathematical expressions or solutions. The box labeled \((x + 2)\) likely serves as a factor for all polynomial expressions connected to it, meaning that the adjacent polynomials can be factored in a way that involves \((x + 2)\).

To solve for each red empty box, we will factor the corresponding polynomials (where applicable) or manipulate the expressions to see how the structure fits into the puzzle.

### Let's start filling the boxes:

#### Empty Box 1 (above \(x^2 + 11x + 18\)):
- Polynomial: \(x^2 + 11x + 18\)
  - This can be factored as \((x + 2)(x + 9)\).
  - **Answer: \(x + 9\)**

#### Empty Box 2 (above \(x^3 + 6x^2 + 11x + 6\)):
- Polynomial: \(x^3 + 6x^2 + 11x + 6\)
  - This is factored as \((x + 2)(x^2 + 4x + 3)\).
  - The quadratic part \(x^2 + 4x + 3\) can further be factored as \((x + 1)(x + 3)\).
  - **Answer: \(x^2 + 4x + 3\)** (or could also be written as \((x + 1)(x + 3)\))

#### Empty Box 3 (next to \(x^2 + 8x + 15\)):
- Polynomial: \(x^2 + 8x + 15\)
  - This can be factored as \((x + 3)(x + 5)\).
  - **Answer: \(x + 3\)**

#### Empty Box 4 (next to \(x^3 + 6x^2 - 25x + 18\)):
- Polynomial: \(x^3 + 6x^2 - 25x + 18\)
  - This can be factored as \((x + 2)(x^2 + 4x - 9)\).
  - **Answer: \(x^2 + 4x - 9\)**

#### Empty Box 5 (between \(x^3 + 6x^2 - 25x + 18\) and \(x^4 - 17x^2 + 16\)):
- We can infer that the common factor is \(x + 2\).
- We'll continue factoring the related expressions.

#### Empty Box 6 (next to \(x^4 - 17x^2 + 16\)):
- Polynomial: \(x^4 - 17x^2 + 16\)
  - This can be factored as \((x^2 - 1)(x^2 - 16)\), which simplifies to \((x - 1)(x + 1)(x - 4)(x + 4)\).
  - **Answer: \(x - 4\)**

#### Empty Box 7 (next to \(x^3 - 2x^2 - 23x + 60\)):
- Polynomial: \(x^3 - 2x^2 - 23x + 60\)
  - This can be factored as \((x + 2)(x^2 - 4x - 30)\), where \(x^2 - 4x - 30\) can be factored further.
  - **Answer: \(x^2 - 4x - 30\)**

#### Empty Box 8 (next to \(x^2 - 7x + 10\)):
- Polynomial: \(x^2 - 7x + 10\)
  - This can be factored as \((x - 5)(x - 2)\).
  - **Answer: \(x - 2\)**

#### Empty Box 9 (below \(x^3 - 7x^2 - 14x + 120\)):
- Polynomial: \(x^3 - 7x^2 - 14x + 120\)
  - This can be factored as \((x + 2)(x^2 - 9x + 60)\).
  - **Answer: \(x^2 - 9x + 60\)**

#### Empty Box 10 (below \(x^2 - 9x + 18\)):
- Polynomial: \(x^2 - 9x + 18\)
  - This can be factored as \((x - 3)(x - 6)\).
  - **Answer: \(x - 6\)**

#### Empty Box 11 (in the center):
- Based on the structure of the puzzle and the fact that \((x + 2)\) appears centrally, the center might require us to find an intersection or consistent factor. It seems logical that this box might involve a factor of \(x + 2\) repeated.
- **Answer: \(x + 2\)**

### Summary of the answers for the empty boxes:
1. \(x + 9\)
2. \(x^2 + 4x + 3\)
3. \(x + 3\)
4. \(x^2 + 4x - 9\)
5. \(x^2 - 9x + 60\)
6. \(x - 4\)
7. \(x^2 - 4x - 30\)
8. \(x - 2\)
9. \(x^2 - 9x + 60\)
10. \(x - 6\)
11. \(x + 2\)

Would you like any details on the factoring process or further clarifications?

### 5 Related Questions:
1. How do you factor a cubic polynomial involving terms like \(x^3 + ax^2 + bx + c\)?
2. Can all quadratic polynomials be factored easily, or are there exceptions?
3. What is the significance of the \(x + 2\) factor in all these polynomials?
4. How can you apply the Remainder Theorem or Synthetic Division in such puzzles?
5. Can higher-degree polynomials (quartic, etc.) always be factored using simple methods?

### Tip:
When factoring polynomials, always check for common factors and apply the quadratic or cubic formulas if necessary.

Fill the 11 red empty boxes in the puzzle using the polynomial factors or related expressions.

Explore how to solve a polynomial factorization puzzle by filling in the 11 red empty boxes using factored expressions like (x + 2) and (x + 9). This exercise covers key algebraic concepts including cubic and quadratic equations, polynomial factorization, and the factor theorem. Suitable for Grades 9-12.

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