https://file.aiquickdraw.com/mathbot/file/1727374511965go19d3nt.jpg

The problem in the image asks to multiply and divide algebraic expressions, then simplify the result. Let's break down the given expressions:

### Expressions:
\[
\frac{(4x^2 + 5x + 1)}{(y^2 - 5y + 4)} \times \frac{(y^2 + 2y - 24)}{(12x^2 + 7x + 1)} \div \frac{(7x^2 + 6x - 1)}{(6x^2 - 7x - 3)}
\]

### Approach:
1. **Division of fractions**: Dividing by a fraction is the same as multiplying by its reciprocal. Thus, we'll multiply the first two terms and the reciprocal of the third term.

2. **Factor each expression**:
    - \( 4x^2 + 5x + 1 \) factors as \( (4x + 1)(x + 1) \)
    - \( y^2 - 5y + 4 \) factors as \( (y - 4)(y - 1) \)
    - \( y^2 + 2y - 24 \) factors as \( (y + 6)(y - 4) \)
    - \( 12x^2 + 7x + 1 \) factors as \( (3x + 1)(4x + 1) \)
    - \( 7x^2 + 6x - 1 \) factors as \( (7x - 1)(x + 1) \)
    - \( 6x^2 - 7x - 3 \) factors as \( (3x + 1)(2x - 3) \)

3. **Simplify by canceling out common factors**:
    - \( 4x + 1 \) cancels out.
    - \( y - 4 \) cancels out.
    - \( x + 1 \) cancels out.
    - \( 3x + 1 \) cancels out.

4. **Final expression**:
    After canceling out, you are left with:
    \[
    \frac{(y + 6)}{(2x - 3)(y - 1)}
    \]

### Simplified Result:
\[
\frac{y + 6}{(2x - 3)(y - 1)}
\]

Let me know if you need more clarification or further details!

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**Follow-up questions:**
1. How do you factor quadratic expressions?
2. What is the rule for dividing by a fraction?
3. How do you cancel terms in algebraic fractions?
4. Can you always simplify an expression by factoring?
5. What are the steps to check if your simplification is correct?

**Tip**: Always check for common factors before multiplying or dividing algebraic expressions. This will help simplify your work early on.

Multiply and divide as indicated, then simplify the result.

This guide explains how to simplify complex algebraic expressions by multiplying and dividing. The problem involves factoring quadratic expressions and applying the multiplication and division of fractions. This tutorial is suitable for high school students learning advanced algebra.

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