If the lengths of adjacent sides of a rectangle are represented by [tex]4x^2 - 3[/tex] and [tex]7x + 2[/tex] units, respectively, for some value of [tex]x[/tex], then which of the following expressions represents the area, in square units, of the rectangle?

A) [tex]11x^2 + 13x - 6[/tex]

B) [tex]28x^3 - 6[/tex]

C) [tex]28x^3 - 13x - 6[/tex]

D) [tex]28x^3 - 8x^2 + 21x - 6[/tex]

E) [tex]28x^3 + 8x^2 - 21x - 6[/tex]

To find the expression that represents the area of the rectangle, we start with the given lengths of the adjacent sides of the rectangle: [tex]$4x^2 - 3$[/tex] and [tex]$7x + 2$[/tex].

The area of a rectangle is calculated by multiplying the lengths of its two adjacent sides. So, we need to multiply these expressions together:

1. Write down the expressions:
- First side: [tex]$4x^2 - 3$[/tex]
- Second side: [tex]$7x + 2$[/tex]

2. Multiply the expressions:

We use the distributive property (often called FOIL when dealing with binomials), which involves multiplying each term from the first expression by each term from the second expression:

[tex]\[
(4x^2 - 3)(7x + 2)
\][/tex]

3. Apply the distributive property:

[tex]\[
= (4x^2 \cdot 7x) + (4x^2 \cdot 2) + (-3 \cdot 7x) + (-3 \cdot 2)
\][/tex]

4. Calculate each term:

- [tex]$4x^2 \cdot 7x = 28x^3$[/tex]
- [tex]$4x^2 \cdot 2 = 8x^2$[/tex]
- [tex]$-3 \cdot 7x = -21x$[/tex]
- [tex]$-3 \cdot 2 = -6$[/tex]

5. Combine all the terms:

[tex]\[
= 28x^3 + 8x^2 - 21x - 6
\][/tex]

So, the expression that represents the area, in square units, of the rectangle is:

[tex]\[28x^3 + 8x^2 - 21x - 6\][/tex]

This matches option (e) from the list of choices provided.