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The problem you have provided is a division of rational expressions:

\[
\frac{3x^2 + 9x - 12}{4x^2 - 8x - 8} \div \frac{x^2 + x - 12}{x^2 + 2x - 15}
\]

To solve this, we can follow these steps:

### Step 1: Factor each polynomial
We will factor the numerators and denominators of both rational expressions.

#### For the first fraction:
\[
3x^2 + 9x - 12 \quad \text{(factor out 3 from all terms)} \Rightarrow 3(x^2 + 3x - 4)
\]
Factor the quadratic expression \(x^2 + 3x - 4\):
\[
x^2 + 3x - 4 = (x + 4)(x - 1)
\]
Thus, the first numerator becomes:
\[
3(x + 4)(x - 1)
\]

Now for the denominator:
\[
4x^2 - 8x - 8 \quad \text{(factor out 4 from all terms)} \Rightarrow 4(x^2 - 2x - 2)
\]
Unfortunately, \(x^2 - 2x - 2\) does not factor nicely over the integers, so we leave it as it is.

#### For the second fraction:
\[
x^2 + x - 12
\]
This can be factored as:
\[
x^2 + x - 12 = (x + 4)(x - 3)
\]

And the denominator:
\[
x^2 + 2x - 15
\]
This can be factored as:
\[
x^2 + 2x - 15 = (x + 5)(x - 3)
\]

### Step 2: Rewrite the expression
Now we have the factored form of both fractions:
\[
\frac{3(x + 4)(x - 1)}{4(x^2 - 2x - 2)} \div \frac{(x + 4)(x - 3)}{(x + 5)(x - 3)}
\]

### Step 3: Multiply by the reciprocal
Dividing by a fraction is the same as multiplying by its reciprocal, so we multiply the first fraction by the reciprocal of the second:
\[
\frac{3(x + 4)(x - 1)}{4(x^2 - 2x - 2)} \times \frac{(x + 5)(x - 3)}{(x + 4)(x - 3)}
\]

### Step 4: Cancel common factors
We can cancel out the common factors in the numerator and denominator:
- The factor \((x + 4)\) appears in both the numerator and denominator, so it cancels out.
- The factor \((x - 3)\) also appears in both, so it cancels out as well.

We are left with:
\[
\frac{3(x - 1)(x + 5)}{4(x^2 - 2x - 2)}
\]

### Final Answer:
The simplified expression is:
\[
\frac{3(x - 1)(x + 5)}{4(x^2 - 2x - 2)}
\]

Let me know if you'd like any further details or have more questions!

---

Here are some related questions:
1. How do you factor quadratic expressions in general?
2. Why do we multiply by the reciprocal when dividing fractions?
3. What happens if a quadratic doesn't factor easily (like \(x^2 - 2x - 2\))?
4. Can you cancel terms before factoring a rational expression?
5. What would happen if you multiplied out the factors again?

**Tip:** When dividing rational expressions, always simplify by factoring first—it makes it easier to cancel terms!

Divide: (3x^2 + 9x - 12) / (4x^2 - 8x - 8) ÷ (x^2 + x - 12) / (x^2 + 2x - 15)

This problem demonstrates how to divide rational expressions by factoring quadratic equations and simplifying. Step-by-step, the expressions are factored and reduced, leading to the simplified result. This is suitable for students in Grades 10-12.

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