Answer :
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.
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.
