The given expression is:

$\frac{(9x^2 - 16)(6x^2 + 17x + 12)(4x^2 + 11x + 6)}{(8x^2 + 26x + 15)(2x^2 + 5x - 3)}$

Step-by-Step Approach:

1. Factor the Quadratics:
   • Let's try factoring the quadratic expressions in both the numerator and the denominator, if possible.

1. Factor $9x^2 - 16$ (difference of squares):

$9x^2 - 16 = (3x - 4)(3x + 4)$

2. Factor $6x^2 + 17x + 12$:

We need two numbers that multiply to $6 \times 12 = 72$ and add up to $17$. These are $9$ and $8$. So, we can factor this as: $6x^2 + 17x + 12 = (2x + 3)(3x + 4)$

3. Factor $4x^2 + 11x + 6$:

We need two numbers that multiply to $4 \times 6 = 24$ and add up to $11$. These are $8$ and $3$. So, we can factor this as: $4x^2 + 11x + 6 = (x + 2)(4x + 3)$

4. Factor $8x^2 + 26x + 15$:

We need two numbers that multiply to $8 \times 15 = 120$ and add up to $26$. These are $10$ and $12$. So, we can factor this as: $8x^2 + 26x + 15 = (4x + 5)(2x + 3)$

5. Factor $2x^2 + 5x - 3$:

We need two numbers that multiply to $2 \times -3 = -6$ and add up to $5$. These are $6$ and $-1$. So, we can factor this as: $2x^2 + 5x - 3 = (2x - 1)(x + 3)$

Step 2: Simplify the Expression

Now substitute all the factored forms into the expression:

$\frac{(3x - 4)(3x + 4)(2x + 3)(3x + 4)(x + 2)(4x + 3)}{(4x + 5)(2x + 3)(2x - 1)(x + 3)}$

Step 3: Cancel Common Terms

• The factor $(3x + 4)$ appears twice in the numerator and once in the denominator, so we can cancel it out.
• The factor $(2x + 3)$ appears once in the numerator and once in the denominator, so we can cancel it out as well.

After canceling, we are left with:

$\frac{(3x - 4)(x + 2)(4x + 3)}{(4x + 5)(2x - 1)(x + 3)}$

Final Simplified Expression:

$\frac{(3x - 4)(x + 2)(4x + 3)}{(4x + 5)(2x - 1)(x + 3)}$

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Would you like further clarification or additional steps on this? Feel free to ask!

Additional Questions:

1. How do you factor different types of quadratic expressions efficiently?
2. What happens if the given expression has a higher degree polynomial—how do we approach factoring?
3. Could the cancellation process be impacted if the expression was in a different form, like involving cubes?
4. How do you check if the factorization is correct?
5. What are some real-life applications where simplifying rational expressions is useful?

Tip:

When factoring quadratics, always check for common patterns like difference of squares, perfect square trinomials, or use factoring by grouping for more complex trinomials.