To simplify the given expression, let's rewrite it step-by-step:

The expression is:

\[
\frac{(4x + 2) \times (5x + 4)}{2} \times \frac{(4x + 2) \times (5x + 4)}{2}
\]

### Step 1: Understanding the expression

Notice that the expression is a product of two identical fractions. To simplify, we can write:

\[
\left( \frac{(4x + 2) \times (5x + 4)}{2} \right)^2
\]

### Step 2: Simplify the expression inside the parentheses

First, let's simplify the product inside the fraction:

\[
(4x + 2) \times (5x + 4)
\]

To expand this, use the distributive property (FOIL method):

\[
(4x + 2)(5x + 4) = 4x \cdot 5x + 4x \cdot 4 + 2 \cdot 5x + 2 \cdot 4
\]

\[
= 20x^2 + 16x + 10x + 8
\]

\[
= 20x^2 + 26x + 8
\]

### Step 3: Substitute back into the fraction

Now, substitute this back into the expression:

\[
\left( \frac{20x^2 + 26x + 8}{2} \right)^2
\]

Simplify inside the parentheses by dividing by 2:

\[
= \left( 10x^2 + 13x + 4 \right)^2
\]

### Step 4: Final expression

Since the expression inside the parentheses is simplified, squaring it gives us the final simplified form:

\[
(10x^2 + 13x + 4)^2
\]

This is the simplified form of the given expression.

### Would you like to see the expanded form of the square, or do you have any other questions?

Here are some related questions that might help you understand better:

1. How do you expand a binomial squared, like \((a + b)^2\)?
2. What is the FOIL method and how is it used in algebra?
3. How do you factor quadratic expressions?
4. What are the common mistakes to avoid when simplifying expressions?
5. How do you solve quadratic equations by factoring?

**Tip:** Always check if a mathematical expression can be simplified or factored to make further calculations easier.

Learn how to simplify the expression (4x + 2)(5x + 4) / 2 with a detailed step-by-step solution. This problem covers algebraic concepts like polynomial multiplication and fraction simplification using the FOIL method. Suitable for students in Grades 9-12.

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