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The given expression is:

\[
\frac{2x^2 + x - 15}{ax + 3a - 2bx - 6b}
\]

To simplify this expression, let's follow these steps:

### Step 1: Simplify the numerator
The numerator is \(2x^2 + x - 15\). Let's try to factor it.

We want two numbers that multiply to \(2 \times -15 = -30\) and add to \(1\) (the coefficient of \(x\)).

The numbers are \(6\) and \(-5\), because \(6 \times -5 = -30\) and \(6 + (-5) = 1\).

Thus, we can rewrite the middle term and factor by grouping:

\[
2x^2 + 6x - 5x - 15 = 2x(x + 3) - 5(x + 3) = (2x - 5)(x + 3)
\]

So, the numerator is factored as:

\[
2x^2 + x - 15 = (2x - 5)(x + 3)
\]

### Step 2: Simplify the denominator
The denominator is \(ax + 3a - 2bx - 6b\). We can factor by grouping terms that have common factors:

\[
ax + 3a - 2bx - 6b = a(x + 3) - 2b(x + 3) = (a - 2b)(x + 3)
\]

### Step 3: Simplify the entire expression
Now we have:

\[
\frac{(2x - 5)(x + 3)}{(a - 2b)(x + 3)}
\]

Since \(x + 3\) appears in both the numerator and denominator, we can cancel it out (as long as \(x \neq -3\), to avoid division by zero):

\[
\frac{2x - 5}{a - 2b}
\]

### Final Answer:
\[
\frac{2x - 5}{a - 2b}
\]

Let me know if you need further details or have any questions.

### Related Questions:
1. How do you factor quadratic expressions in general?
2. What is factoring by grouping, and when is it useful?
3. How can you simplify algebraic fractions?
4. What happens if \(x = -3\) in this expression?
5. How would the expression change if there was no common factor to cancel out?

### Tip:
Always check for common factors in both the numerator and denominator when simplifying algebraic fractions!

Simplify: (2x^2 + x - 15) / (ax + 3a - 2bx - 6b)

Learn how to simplify a rational expression by factoring the quadratic in the numerator and grouping terms in the denominator. This detailed solution walks you through key algebraic concepts for Grades 9-12.

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