The problem asks for the area of a rectangle, where the length is represented by $x^2 + 3x + 2$ and the width is represented by $4x$. The area $A$ of a rectangle is given by the formula:

$A = \text{length} \times \text{width}$

Substituting the given expressions:

$A = (x^2 + 3x + 2) \times 4x$

Now, distribute $4x$ to each term inside the parentheses:

$A = 4x(x^2) + 4x(3x) + 4x(2)$

This simplifies as follows:

$A = 4x^3 + 12x^2 + 8x$

Thus, the area of the rectangle is:

$4x^3 + 12x^2 + 8x$

The correct option is the first one: $4x^3 + 12x^2 + 8x$.

Would you like more details or have any further questions?

Here are five related questions to deepen your understanding:

1. What is the general formula for the area of a rectangle?
2. How would the area change if the width was represented by a constant rather than a variable?
3. Can you factor the polynomial expression for the area of this rectangle?
4. How does distributing work when multiplying polynomials?
5. What happens to the area when both length and width increase?

Tip: Always double-check your multiplication when working with polynomials to avoid simple mistakes in the expansion process.