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The image contains a list of problems involving the combination of similar terms and operations like FOIL (First, Outside, Inside, Last) for expanding binomials.

Let's go through each problem:

### Problem 1: 
\[
(2x^2 - 3x^2 - 5x) + (3x^2 - 5x^2 - 8x)
\]

**Step 1:** Combine like terms inside the parentheses:
- \(2x^2 - 3x^2 = -x^2\)
- \(3x^2 - 5x^2 = -2x^2\)
- \(-5x\) and \(-8x\) remain as they are.

**Step 2:** Simplify further:
\[
(-x^2 - 5x) + (-2x^2 - 8x)
\]
Combine like terms across the two expressions:
\[
(-x^2 - 2x^2) + (-5x - 8x) = -3x^2 - 13x
\]
So, the simplified expression is:
\[
\boxed{-3x^2 - 13x}
\]

### Problem 2:
\[
(3x + 2)(2x^2 - 5x + 4)
\]

**Step 1:** Apply the FOIL method:
- **First:** \(3x \times 2x^2 = 6x^3\)
- **Outside:** \(3x \times -5x = -15x^2\)
- **Inside:** \(2 \times 2x^2 = 4x^2\)
- **Last:** \(2 \times -5x = -10x\)

Combine like terms:
\[
6x^3 + (-15x^2 + 4x^2) - 10x + 8 = 6x^3 - 11x^2 - 10x + 8
\]
So, the simplified expression is:
\[
\boxed{6x^3 - 11x^2 - 10x + 8}
\]

### Problem 3:
\[
\frac{2a^2b}{3c} + \frac{5a^2b^2}{6c^2}
\]

This is a sum of fractions with different denominators, so we'll try to find a common denominator:
- The common denominator would be \(6c^2\).

Rewrite the first term:
\[
\frac{2a^2b \cdot 2c}{3c \cdot 2c} = \frac{4a^2bc}{6c^2}
\]

Now, the expression becomes:
\[
\frac{4a^2bc}{6c^2} + \frac{5a^2b^2}{6c^2} = \frac{4a^2bc + 5a^2b^2}{6c^2}
\]
So, the simplified expression is:
\[
\boxed{\frac{4a^2bc + 5a^2b^2}{6c^2}}
\]

### Problem 4:
\[
\frac{2x^2 - x - 2}{2x - 2}
\]

Factor both the numerator and denominator:
- Numerator: \(2x^2 - x - 2 = (2x + 1)(x - 2)\)
- Denominator: \(2x - 2 = 2(x - 1)\)

Since there are no common factors to cancel out, the expression remains:
\[
\boxed{\frac{(2x + 1)(x - 2)}{2(x - 1)}}
\]

### Problem 5:
\[
\frac{(x^2 - 4)(x^2 - 9)}{x - 2}
\]

Apply the difference of squares for both terms in the numerator:
- \(x^2 - 4 = (x - 2)(x + 2)\)
- \(x^2 - 9 = (x - 3)(x + 3)\)

So, the expression becomes:
\[
\frac{(x - 2)(x + 2)(x - 3)(x + 3)}{x - 2}
\]
You can cancel out the \(x - 2\) term:
\[
\boxed{(x + 2)(x - 3)(x + 3)}
\]

---

If you have any questions or need further clarification, let me know! Here are some follow-up questions:

1. What is the FOIL method, and how does it apply to binomials?
2. How can you identify when to apply the difference of squares?
3. What are common mistakes when combining like terms?
4. How do you factor quadratic expressions like in problem 4?
5. How can you simplify fractions with different denominators effectively?

**Tip:** Always look for common factors or patterns like the difference of squares when simplifying algebraic expressions!

Explore how to use the FOIL method to expand binomials effectively. This detailed guide includes step-by-step solutions to problems involving binomial expansion and fraction operations in algebra. Suitable for Grades 9-12.

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