https://file.aiquickdraw.com/mathbot/file/1725687797399xes8n5ev.jpg

The image contains several algebraic expressions that need to be simplified. Let's go through the solutions one by one:

### 1) \(\frac{3y}{4y} + \frac{2x}{4y}\)
- Combine the terms over a common denominator:
\[
\frac{3y + 2x}{4y}
\]
So, the answer is **D) \(\frac{3y + 2x}{4y}\)**.

### 2) \(\frac{3x}{y} - \frac{2}{3y^2}\)
- Since the terms already have different denominators, they can't be combined further. This is the simplified form.
The answer is **A) \(\frac{3x}{y} - \frac{2}{3y^2}\)**.

### 3) \(\frac{x}{y^2} - \frac{y}{2}\)
- Combine over the common denominator:
\[
\frac{2x - y^3}{2y^2}
\]
The answer is **B) \(\frac{2x - y^3}{2y^2}\)**.

### 4) \(\frac{12x^2 - 4xy - 15x}{30y}\)
- Factor the numerator:
\[
\frac{x(12x - 4y - 15)}{30y}
\]
This expression cannot be simplified further.
The answer is **C) \(\frac{x(12x - 4y - 15)}{30y}\)**.

### 5) \(\frac{4x}{3y} + \frac{2x}{3y^2}\)
- The terms already have a common denominator:
\[
\frac{4xy + 2x}{3y^2}
\]
The answer is **B) \(\frac{4xy + 2x}{3y^2}\)**.

### 6) \(\frac{6a + 3}{4b^2} + \frac{3a}{2b^2}\)
- Combine over a common denominator:
\[
\frac{12a + 6 + 6a}{4b^2} = \frac{18a + 6}{4b^2}
\]
The answer is **A) \(\frac{18a + 6}{4b^2}\)**.

### 7) \(\frac{5n - 1}{4} + \frac{3n}{6}\)
- Combine over a common denominator:
\[
\frac{30n - 6 + 12n}{24} = \frac{42n - 6}{24}
\]
The answer is **A) \(\frac{42n - 6}{24}\)**.

### 8) \(\frac{3b + 2c}{5b} - \frac{c}{b^2}\)
- Simplify and combine over the common denominator:
\[
\frac{3b^2 + 2cb - 5c}{5b^2}
\]
The answer is **D) \(\frac{3b^2 + 2cb - 5c}{5b^2}\)**.

Would you like more detailed steps for any of these solutions or explanations of the techniques used?

Here are some related questions you might consider:
1. How do you factor complex polynomials like the ones in problem 4?
2. How do you determine the least common denominator when adding rational expressions?
3. How do you know when an algebraic expression is fully simplified?
4. What are common strategies for dealing with variables in both the numerator and denominator?
5. How can you simplify rational expressions involving quadratic terms?

**Tip**: Always check for common factors in both the numerator and denominator before combining fractions. It simplifies the process and reduces errors!

Simplify the following algebraic expressions, including combining like terms and working with fractions.

Learn how to simplify algebraic expressions with step-by-step solutions, including combining like terms, factoring, and working with fractions. Suitable for students in Grades 9-11.

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