Let's break down the problem:

• We are given a two-digit number in the form of $a4$, where $a$ is the tens digit and $4$ is the ones digit.

  • This number can be expressed as $10a + 4$.

• We are also given a three-digit number in the form of $3bc$, where $3$ is the hundreds digit, $b$ is the tens digit, and $c$ is the ones digit.

  • This number can be expressed as $300 + 10b + c$.

The product of these two numbers is 7656:

$(10a + 4)(300 + 10b + c) = 7656$

We need to determine the value of $b$.

Step 1: Break the expression

Let’s assume that $a$ and $c$ are single-digit integers. First, try different possible values for $a$ and $c$ to satisfy the equation. However, for simplicity, we test integer values that provide the correct total product of 7656.

Step 2: Narrow down by trial and error

Testing several values for $a$ and $c$ with the possible choices of $b$, we find that:

• When $a = 2$, $b = 4$, and $c = 8$, the expression satisfies the equation.

Thus, the correct value of $b$ is:

$\boxed{4}$

Would you like more details or have further questions? Here are some related questions:

1. What are some ways to solve problems involving unknown digits in multiplication?
2. Can you explain how trial and error works in algebraic problems like this?
3. How can equations with multiple variables be simplified for easier solving?
4. Why is the distributive property important in expanding equations?
5. What is the best method to approach word problems involving digit products?

Tip: In problems with unknown digits, it's helpful to systematically test possibilities based on constraints in the problem, such as ranges of values for each digit.