Problem:
The product of two alternate odd integers exceeds three times the smaller by 12. What is the larger number? Explain in detail.

Solution:

Step 1: Understand the problem
Let's break down the problem and identify the key information given:
- We are looking for two alternate odd integers.
- The product of these two integers exceeds three times the smaller by 12.

Step 2: Formulate equations
To solve this problem, we need to set up equations based on the given information.

Let's assume the two alternate odd integers as (2n+1) and (2n+3), where n is any positive integer.
- (2n+1) represents the smaller odd integer.
- (2n+3) represents the larger odd integer.

According to the problem, the product of these two integers exceeds three times the smaller integer by 12. Mathematically, we can represent this as:
(2n+1) * (2n+3) = 3(2n+1) + 12

Step 3: Solve the equation
Let's expand and simplify the equation:
4n^2 + 8n + 3 = 6n + 3 + 12
4n^2 + 8n + 3 = 6n + 15

Rearranging the equation:
4n^2 + 2n - 12 = 0

We can solve this quadratic equation by factoring or using the quadratic formula. Factoring this equation, we get:
(2n - 3)(2n + 4) = 0

Setting each factor equal to zero and solving for n, we find two possible values for n:
1. 2n - 3 = 0 ---> 2n = 3 ---> n = 3/2 (not a positive integer)
2. 2n + 4 = 0 ---> 2n = -4 ---> n = -2 (not a positive integer)

Since n must be a positive integer, we cannot consider the above solutions.

Step 4: Find the larger number
Since we couldn't find a valid solution for n, it means that there are no two alternate odd integers satisfying the given conditions. Therefore, there is no larger number in this case.

Conclusion:
Based on the given conditions, there is no larger number that satisfies the problem statement.