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The image provided contains a mathematical exercise asking to apply factorization by grouping for polynomial expressions. Below are the solutions to the points a, b, and c based on what is visible in the image.

### a) \( 3x + 6y + ax + 2ay \)

**Step 1: Group the terms.**
\[
(3x + ax) + (6y + 2ay)
\]

**Step 2: Factor out the common factors from each group.**
\[
x(3 + a) + 2y(3 + a)
\]

**Step 3: Factor out the common binomial.**
\[
(3 + a)(x + 2y)
\]

So, the factored form is:
\[
\boxed{(3 + a)(x + 2y)}
\]

### b) \( xy - x - 2y + 2 \)

**Step 1: Group the terms.**
\[
(xy - x) - (2y - 2)
\]

**Step 2: Factor out the common factors from each group.**
\[
x(y - 1) - 2(y - 1)
\]

**Step 3: Factor out the common binomial.**
\[
(y - 1)(x - 2)
\]

So, the factored form is:
\[
\boxed{(y - 1)(x - 2)}
\]

### c) \( x^2 + 5x + 6 \)

**Step 1: Identify the factors of 6 that add up to 5.**
\[
2 \times 3 = 6 \quad \text{and} \quad 2 + 3 = 5
\]

**Step 2: Write the middle term as the sum of two terms.**
\[
x^2 + 2x + 3x + 6
\]

**Step 3: Group the terms and factor.**
\[
(x^2 + 2x) + (3x + 6)
\]

**Step 4: Factor out the common factors from each group.**
\[
x(x + 2) + 3(x + 2)
\]

**Step 5: Factor out the common binomial.**
\[
(x + 2)(x + 3)
\]

So, the factored form is:
\[
\boxed{(x + 2)(x + 3)}
\]

Would you like more details on any of these steps, or do you have any other questions?

### Related Questions:
1. What are the different methods for factorizing polynomials?
2. How can we factor trinomials where the leading coefficient is not 1?
3. What is the significance of the common binomial factor in grouping?
4. Can you explain how to factor higher-degree polynomials by grouping?
5. How do you know when to use the factorization by grouping method?

### Tip:
When dealing with polynomials, always check for a common factor first before attempting to use more complex factorization methods.

Understand how to apply factorization by grouping to solve polynomial expressions. This exercise provides step-by-step solutions for expressions involving terms like 3x + 6y + ax + 2ay, xy - x - 2y + 2, and quadratic expressions such as x^2 + 5x + 6. Suitable for students in Grades 8-10.

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