To solve this problem quickly, you need to follow the steps for dividing two rational expressions, which essentially involves multiplying by the reciprocal of the second fraction.

Here’s the expression you provided:

$\frac{2x^2 - 5x - 3}{x^2 + 3x - 10} \div \frac{x^2 - 9}{2x^2 + 16x + 30}$

Steps:

1. Factor each part of the expression:

   • Factor both the numerators and denominators of each fraction.

   First fraction:

   • $2x^2 - 5x - 3$ factors as $(2x + 1)(x - 3)$.
   • $x^2 + 3x - 10$ factors as $(x + 5)(x - 2)$.

   Second fraction:

   • $x^2 - 9$ is a difference of squares, so it factors as $(x - 3)(x + 3)$.
   • $2x^2 + 16x + 30$ factors as $2(x + 3)(x + 5)$.

   So the expression becomes:

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

2. Multiply by the reciprocal:

   • Division of fractions is the same as multiplying by the reciprocal. So the expression becomes:

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

3. Cancel common factors:

   • Cancel out $(x - 3)$ from the numerator and denominator.
   • Cancel out $(x + 5)$ from the numerator and denominator.
   • Cancel out $(x + 3)$ from the numerator and denominator.

   After cancellation, you're left with:

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

4. Final answer: The simplified expression is:

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

Key tip:

When solving rational expressions like this, always factor both numerators and denominators first and look for common factors to cancel. This will save time and simplify the problem.

Would you like more details on any of the steps, or do you have any questions?

Here are 5 related questions to further practice:

1. How do you factor quadratic expressions quickly?
2. What is the rule for multiplying rational expressions?
3. How can you spot common factors easily in rational expressions?
4. What are the key strategies for dividing fractions in algebra?
5. How do you deal with special factorizations like the difference of squares?

Tip: Always rewrite division of fractions as multiplication by the reciprocal before you start simplifying.