$$ \frac{1}{2x^2+x-15}\text{ divide by } $$$$ \frac{1}{3x^2+9x} $$ simplify to $$ \frac{ax}{bx+c} $$where a,b and c are integers to be found

We are asked to divide two rational expressions and simplify the result into the form:

\[
\frac{ax}{bx+c}
\]

### Step 1: Set up the division
We are given the following expression:

\[
\frac{1}{2x^2 + x - 15} \div \frac{1}{3x^2 + 9x}
\]

Dividing by a fraction is the same as multiplying by its reciprocal. So we can rewrite this as:

\[
\frac{1}{2x^2 + x - 15} \times \frac{3x^2 + 9x}{1}
\]

This simplifies to:

\[
\frac{3x^2 + 9x}{2x^2 + x - 15}
\]

### Step 2: Factor both the numerator and the denominator
We now factor both the numerator and the denominator.

- **Numerator:** $ 3x^2 + 9x $ has a common factor of 3:

\[
3x^2 + 9x = 3x(x + 3)
\]

- **Denominator:** $ 2x^2 + x - 15 $ is a quadratic trinomial. We need to factor it:

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

### Step 3: Simplify the expression
Now we substitute the factored forms into the expression:

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

We can cancel out the common factor $ (x + 3) $ in both the numerator and the denominator:

\[
\frac{3x}{2x - 5}
\]

### Step 4: Write in the form $ \frac{ax}{bx + c} $
The simplified form is:

\[
\frac{3x}{2x - 5}
\]

Thus, comparing with $ \frac{ax}{bx + c} $, we have:

- $ a = 3 $
- $ b = 2 $
- $ c = -5 $

### Final answer:
\[
\boxed{\frac{3x}{2x - 5}}
\]

Let me know if you'd like more details or have any further questions! Here are 5 related questions:

1. How do you factor trinomials like $2x^2 + x - 15$?
2. What are some techniques to simplify rational expressions?
3. How can you determine common factors in polynomials?
4. How does dividing by a fraction relate to multiplying by the reciprocal?
5. What steps can you follow to cancel common terms in fractions?

**Tip:** Always factor both the numerator and denominator to check for common factors when simplifying rational expressions!

$$ \frac{1}{2x^2+x-15}\text{ divide by } \frac{1}{3x^2+9x} $$ simplify to $$ \frac{ax}{bx+c} $$ where a, b, and c are integers to be found

Learn how to simplify the expression $$ \frac{1}{2x^2+x-15} \div \frac{1}{3x^2+9x} $$ by factoring and division of rational expressions. This problem explores key algebraic techniques, including factoring quadratic expressions and simplifying fractions. Suitable for students in Grades 9-12.

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