Let the two consecutive odd numbers be represented as $x$ and $x + 2$, where $x$ is the smaller odd number.

The problem states that their product is 2 more than 13, which can be written as:

$x(x + 2) = 13 + 2$

Simplify the right-hand side:

$x(x + 2) = 15$

Now expand the left-hand side:

$x^2 + 2x = 15$

To form a quadratic equation, subtract 15 from both sides:

$x^2 + 2x - 15 = 0$

Next, factor the quadratic equation:

$(x + 5)(x - 3) = 0$

Setting each factor equal to zero gives:

$x + 5 = 0 \quad \text{or} \quad x - 3 = 0$

Solving these:

$x = -5 \quad \text{or} \quad x = 3$

Since we are looking for the greater number, we choose $x = 3$. Thus, the two consecutive odd numbers are $3$ and $5$, and the greater number is:

$\boxed{5}$

Would you like more details on this problem, or have any further questions? Here are five related questions to explore:

1. What is the product of two consecutive even numbers and how does it relate to the sum of 13?
2. How can you solve a similar problem involving three consecutive odd numbers?
3. What are some other methods for factoring quadratic equations besides trial and error?
4. How do you solve quadratic equations using the quadratic formula?
5. How do you determine the roots of a quadratic equation when factoring isn't possible?

Tip: Always check your solutions by substituting them back into the original equation to verify they satisfy the problem.