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It looks like you've uploaded an image containing a series of algebraic fraction expressions that need to be simplified fully. I'll start by solving a couple of the problems for you, then you can let me know if you'd like me to continue with others or dive into any specific one.

Let's begin with **Q1** and **Q2**:

### **Q1:**
\[
\frac{(x+3)(x-5)}{x-6} \times \frac{3x(x-6)}{x-5}
\]

1. First, simplify by canceling out terms common in the numerator and the denominator:
   - The \(x - 6\) terms cancel out.
   - The \(x - 5\) terms cancel out.
   
2. This leaves:
\[
(x+3) \times 3x = 3x(x+3)
\]

Thus, the simplified form is:
\[
3x(x+3) = 3x^2 + 9x
\]

---

### **Q2:**
\[
\frac{10(5x+3)}{2x+3} \times \frac{(2x+1)(2x+3)}{(5x-3)(5x+3)}
\]

1. Cancel out common terms:
   - The \(5x+3\) terms cancel out.
   - The \(2x+3\) terms cancel out.

2. The expression simplifies to:
\[
\frac{10}{1} \times \frac{2x+1}{5x-3} = \frac{10(2x+1)}{5x-3}
\]

Thus, the simplified form is:
\[
\frac{10(2x+1)}{5x-3}
\]

---

Would you like to go over more problems or dive deeper into any steps?

Here are 5 related questions that could deepen your understanding:
1. How do you cancel common factors in a fraction?
2. What happens when variables in the numerator and denominator share a common factor?
3. How do you factor quadratic expressions in more complex problems?
4. Can you always cancel terms directly, or are there restrictions?
5. What is the difference between simplifying and factoring algebraic expressions?

**Tip:** Always factorize the terms first before canceling common factors; this helps avoid errors during simplification!

Simplify the following algebraic fractions: (x+3)(x-5)/(x-6) * 3x(x-6)/(x-5) and 10(5x+3)/(2x+3) * (2x+1)(2x+3)/(5x-3)(5x+3)

Learn how to simplify algebraic fractions step-by-step. This problem covers key algebraic concepts, including simplifying rational expressions by canceling common factors. Suitable for high school students (Grades 9-12).

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