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!