https://r2.erweima.ai/mathbot/image/1725857255351atvnco0q.pdf

The file you uploaded contains a set of mathematical exercises focused on **factoring, simplification, and algebraic manipulation**. Below is a summary of the exercises:

1. Simplify:
   \[
   \frac{4x^2 - 9}{2x^2 + 7x + 6} \times \frac{4x^4 + 6x^3 + 9x^2}{27x^4 + 8x}
   \]
   
2. Simplify:
   \[
   \frac{a^3 - 8}{a^2 - 4} \div \frac{a}{a^3 + 8}
   \]
   
3. Simplify:
   \[
   \frac{2x + 6}{x^2 + 6x + 9} + \frac{5x}{x^2 - 9} + \frac{7}{x - 3}
   \]
   
4. Simplify the following expression:
   \[
   \frac{3}{w - \frac{6}{2w + 1}} \times \frac{5}{w - \frac{8}{2w + 1}}
   \]

5. Simplify:
   \[
   \frac{2ac + bc - 6ad - 3bd}{6ac + 2ad + 3bc + bd}
   \]
   
6. Simplify:
   \[
   \frac{6x^2 - 5x - 6}{x^2 - 4} \div \frac{2x^2 - 3x}{x + 2}
   \]

7. Simplify:
   \[
   \frac{x}{x + 2} - \frac{4}{x - 2} - \frac{x - 3}{6}
   \]

8. Simplify the following:
   \[
   \frac{5}{2x + 2h + 3} - \frac{5}{2x + 3h}
   \]

9. Simplify:
   \[
   \frac{(x + h)^2 - 3(x + h) - (x^2 - 3x)}{h}
   \]

10. Simplify the derivative-related expression:
   \[
   (x^2 + 9)^{4/3} (x + 6)^{-4/3} + (x + 6)^{-1/3} (4(x^2 + 9)^{3/3} (2x))
   \]

Would you like help solving any specific exercise? Let me know if you'd like step-by-step solutions.

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Here are five related questions:

1. How do you factor polynomials?
2. What is the process for simplifying complex fractions?
3. How do you simplify expressions involving exponents and roots?
4. What is the distributive property in algebra?
5. How do you handle division of rational expressions?

**Tip**: When simplifying expressions, always start by factoring as much as possible—factoring makes cancellations easier!

Simplify the following mathematical expressions from the file, which include algebraic manipulation, factoring, and simplification tasks.

Learn to simplify complex algebraic expressions involving factoring, rational expressions, polynomial division, and exponent manipulation. This guide covers various exercises, including simplifying expressions with multiple terms, factoring polynomials, and working with rational expressions. Suitable for students in Grades 10-12.

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