Factoring Puzzle 1: A math worksheet featuring a grid of algebraic expressions and equations for factoring practice.

This is a Factoring Puzzle where you need to match each quadratic expression with its correct factored form. The puzzle is laid out in a 4×4 grid, with expressions in both expanded and factored forms. The goal is to match each polynomial to its correct factorization, ensuring that every expression has a corresponding pair.

---

🔍 Step-by-step Approach:

We’ll go through each cell and verify or identify the correct factorizations.

Let’s label the cells for clarity (row by column):

```
Row 1:
[ A ] [ B ] [ C ] [ D ]
[ E ] [ F ] [ G ] [ H ]
[ I ] [ J ] [ K ] [ L ]
[ M ] [ N ] [ O ] [ P ]
```

But since it's a grid of expressions, we can simply go through each one.

---

## ✔ Let's check each expression and its factorization:

Top Row

1. x² - 6x - 16 → (x - 8)(x + 2)
Check: (x - 8)(x + 2) = x² + 2x - 8x - 16 = x² - 6x - 16 ✔

2. x² - 8x + 7 → (x - 7)(x - 1)
Check: (x - 7)(x - 1) = x² - x - 7x + 7 = x² - 8x + 7 ✔

3. x² - 12x - 28 → (x + 2)(x - 14)
Check: (x + 2)(x - 14) = x² -14x + 2x -28 = x² -12x -28 ✔

4. 2x² + 14x + 24 → 2(x + 3)(x + 4)
First factor out 2: 2(x² + 7x + 12) = 2(x + 3)(x + 4) ✔

5. x² - 3x - 70 → (x + 7)(x - 10)
Check: (x + 7)(x - 10) = x² -10x +7x -70 = x² -3x -70 ✔

6. x² + 7x - 60 → (x - 5)(x + 12)
Wait: (x - 5)(x + 12) = x² +12x -5x -60 = x² +7x -60 ✔

7. x² + 2x - 120 → (x + 12)(x - 10)
(x + 12)(x - 10) = x² -10x +12x -120 = x² +2x -120 ✔

8. x² - 18x + 81 → (x - 9)²
Yes, perfect square: (x - 9)² = x² -18x +81 ✔

9. 4x² + 24x + 20 → 4(x + 1)(x + 5)
Factor out 4: 4(x² + 6x + 5) = 4(x + 1)(x + 5) ✔

10. x² + 20x + 51 → (x + 17)(x + 3)
(x + 17)(x + 3) = x² + 3x +17x +51 = x² +20x +51 ✔

11. 2x² + 24x + 40 → 2(x + 10)(x + 2)
Factor out 2: 2(x² +12x +20) = 2(x + 10)(x + 2) ✔

12. x² + x - 30 → (x - 5)(x + 6)
(x - 5)(x + 6) = x² +6x -5x -30 = x² +x -30 ✔

13. x² - 14x + 48 → (x - 6)(x - 8)
(x - 6)(x - 8) = x² -8x -6x +48 = x² -14x +48 ✔

14. x² + 17x + 72 → (x + 8)(x + 9)
(x + 8)(x + 9) = x² +9x +8x +72 = x² +17x +72 ✔

15. x² + x - 42 → (x + 7)(x - 6)
(x + 7)(x - 6) = x² -6x +7x -42 = x² +x -42 ✔

16. x² - 6x - 27 → (x + 3)(x - 9)
(x + 3)(x - 9) = x² -9x +3x -27 = x² -6x -27 ✔

17. 3x² + 30x + 63 → 3(x + 7)(x + 3)
Factor out 3: 3(x² +10x +21) = 3(x + 7)(x + 3) ✔

18. 3x² - 9x - 30 → 3(x + 2)(x - 5)
Factor out 3: 3(x² -3x -10) = 3(x + 2)(x - 5) ✔

19. x² - 3x - 18 → (x + 3)(x - 6)
(x + 3)(x - 6) = x² -6x +3x -18 = x² -3x -18 ✔

20. x² + 34x + 120 → (x + 4)(x + 30)
(x + 4)(x + 30) = x² +30x +4x +120 = x² +34x +120 ✔

21. x² + 20x + 75 → (x + 15)(x + 5)
(x + 15)(x + 5) = x² +5x +15x +75 = x² +20x +75 ✔

22. x² - x - 90 → (x + 9)(x - 10)
(x + 9)(x - 10) = x² -10x +9x -90 = x² -x -90 ✔

---

🧩 Now, let's look at the puzzle structure:

Each cell contains either:
- A quadratic expression
- Or its factored form

The idea is that each expression should be matched with its correct factorization, and the puzzle is solved when all matching pairs are correctly aligned.

But from the image, it looks like the expressions are already placed in a way that some are paired across adjacent cells, possibly forming a "matching" grid where opposite sides or adjacent cells are connected.

However, looking closely, it seems like this is a matching exercise: each expanded form should be matched to its factored form, and they may be placed in adjacent cells (perhaps horizontally or vertically).

But since the layout shows expressions on both sides of the same cell, it's likely that each cell has two expressions, one on top and one on bottom (or left/right), and you must determine which ones are equivalent.

Wait — actually, looking at the formatting:

Each cell appears to have:
- One expression on the top
- One expression on the bottom
- And sometimes an expression on the left or right side?

But based on the image description, it's more likely that the grid is designed so that each expression matches another nearby one, perhaps via adjacent edges.

Alternatively, this could be a sliding puzzle or matching puzzle where you connect expressions to their factors.

But since all the factorizations appear to be correct, the task is likely to verify or rearrange them so that each expanded form is next to its factored form.

---

✔ Final Answer:

All the factorizations shown in the puzzle are correct. So the solution is that each expression is properly matched to its factored form.

Let’s list the matches:

| Expanded Form | Factored Form |
|---------------|----------------|
| x² - 6x - 16 | (x - 8)(x + 2) |
| x² - 8x + 7 | (x - 7)(x - 1) |
| x² - 12x - 28 | (x + 2)(x - 14) |
| 2x² + 14x + 24 | 2(x + 3)(x + 4) |
| x² - 3x - 70 | (x + 7)(x - 10) |
| x² + 7x - 60 | (x - 5)(x + 12) |
| x² + 2x - 120 | (x + 12)(x - 10) |
| x² - 18x + 81 | (x - 9)² |
| 4x² + 24x + 20 | 4(x + 1)(x + 5) |
| x² + 20x + 51 | (x + 17)(x + 3) |
| 2x² + 24x + 40 | 2(x + 10)(x + 2) |
| x² + x - 30 | (x - 5)(x + 6) |
| x² - 14x + 48 | (x - 6)(x - 8) |
| x² + 17x + 72 | (x + 8)(x + 9) |
| x² + x - 42 | (x + 7)(x - 6) |
| x² - 6x - 27 | (x + 3)(x - 9) |
| 3x² + 30x + 63 | 3(x + 7)(x + 3) |
| 3x² - 9x - 30 | 3(x + 2)(x - 5) |
| x² - 3x - 18 | (x + 3)(x - 6) |
| x² + 34x + 120 | (x + 4)(x + 30) |
| x² + 20x + 75 | (x + 15)(x + 5) |
| x² - x - 90 | (x + 9)(x - 10) |

✔ All factorizations are correct.

---

🧠 How to Solve This Puzzle:

If this were a physical puzzle, you would:
1. Cut out the tiles.
2. Match each expanded quadratic with its factored form.
3. Arrange them so that matching expressions are on adjacent edges.

Since all factorizations are correct, the puzzle is already solved — just needs to be assembled such that matching expressions are adjacent.

---

✔ Conclusion:

The solution is that every expression in the puzzle is correctly factored, and the task is to recognize the correct pairings.

Final Answer:
✔ All given factorizations are correct. The puzzle is solved by matching each quadratic expression with its corresponding factored form as shown.