Math worksheet with binomial multiplication problems and a riddle to solve.

To solve the problem, we need to factor each quadratic expression in the grid and match them with the binomial factors provided. By shading the correct areas containing the answers, we will uncover the answer to the riddle: "What did the lawyer name his daughter?"

Step-by-Step Solution:

#### 1. Factor each quadratic expression in the grid:
We will factor each quadratic expression row by row.

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Row 1:
1. \( 18x^2 + 34x + 4 \)
- Factors: \( (9x + 2)(2x + 2) \)
- Simplify: \( (9x + 2)(2)(x + 1) \)
- Final factorization: \( (9x + 2)(x + 1) \)

2. \( x^2 - x - 1 \)
- This quadratic does not factor nicely over the integers.
- Final factorization: Not factorable over integers.

3. \( 3x^2 + 4x + 1 \)
- Factors: \( (3x + 1)(x + 1) \)

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Row 2:
4. \( 5x^2 - 22x - 15 \)
- Factors: \( (5x + 3)(x - 5) \)

5. \( 2x^2 + 13x + 12 \)
- Factors: \( (2x + 3)(x + 4) \)

6. \( 12x^2 - 25x + 12 \)
- Factors: \( (3x - 4)(4x - 3) \)

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Row 3:
7. \( 5x^2 - 2x + 21 \)
- This quadratic does not factor nicely over the integers.
- Final factorization: Not factorable over integers.

8. \( 18x^2 - 18x \)
- Factor out the GCF: \( 18x(x - 1) \)
- Final factorization: \( 18x(x - 1) \)

9. \( 25x^2 - 5x - 30 \)
- Factor out the GCF: \( 5(5x^2 - x - 6) \)
- Factor further: \( 5(5x + 5)(x - 1) \)
- Simplify: \( 5(5)(x + 1)(x - 1) \)
- Final factorization: \( 25(x + 1)(x - 1) \)

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Row 4:
10. \( 12x^2 - 8x + 7 \)
- This quadratic does not factor nicely over the integers.
- Final factorization: Not factorable over integers.

11. \( 3x^2 - 3x - 3 \)
- Factor out the GCF: \( 3(x^2 - x - 1) \)
- The quadratic inside does not factor nicely over the integers.
- Final factorization: \( 3(x^2 - x - 1) \)

12. \( x^2 - 8x + 7 \)
- Factors: \( (x - 7)(x - 1) \)

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#### 2. Match the factorizations with the given binomials:
Now, we match the factorizations from the grid with the binomials provided in the list.

- Row 1:
1. \( 18x^2 + 34x + 4 \) → \( (9x + 2)(x + 1) \) → Matches with \( (9x + 2)(x + 1) \) (Option ①).
2. \( x^2 - x - 1 \) → Not factorable.
3. \( 3x^2 + 4x + 1 \) → \( (3x + 1)(x + 1) \) → Matches with \( (3x + 1)(x + 1) \) (Option ③).

- Row 2:
4. \( 5x^2 - 22x - 15 \) → \( (5x + 3)(x - 5) \) → Matches with \( (5x + 3)(x - 5) \) (Option ④).
5. \( 2x^2 + 13x + 12 \) → \( (2x + 3)(x + 4) \) → Matches with \( (2x + 3)(x + 4) \) (Option ⑤).
6. \( 12x^2 - 25x + 12 \) → \( (3x - 4)(4x - 3) \) → Matches with \( (3x - 4)(4x - 3) \) (Option ⑥).

- Row 3:
7. \( 5x^2 - 2x + 21 \) → Not factorable.
8. \( 18x^2 - 18x \) → \( 18x(x - 1) \) → Matches with \( 18x(x - 1) \) (Option ⑦).
9. \( 25x^2 - 5x - 30 \) → \( 25(x + 1)(x - 1) \) → Matches with \( 25(x + 1)(x - 1) \) (Option ⑧).

- Row 4:
10. \( 12x^2 - 8x + 7 \) → Not factorable.
11. \( 3x^2 - 3x - 3 \) → \( 3(x^2 - x - 1) \) → Not factorable.
12. \( x^2 - 8x + 7 \) → \( (x - 7)(x - 1) \) → Matches with \( (x - 7)(x - 1) \) (Option ⑨).

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#### 3. Shade the correct areas:
Shade the areas corresponding to the matched binomials:
- Row 1: Columns 1 and 3
- Row 2: Columns 1, 2, and 3
- Row 3: Columns 2 and 3
- Row 4: Column 3

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#### 4. Read the shaded letters:
The shaded areas spell out the answer to the riddle. Reading the letters in the shaded areas, we get: "ELSA".

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Final Answer:

$$
\boxed{ELSA}
$$