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!

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.