Answer :
To find the quadratic expression that represents the product of the factors [tex]\((2x + 5)(7 - 4x)\)[/tex], follow these steps:
### Step 1: Apply the Distributive Property
We need to multiply each term in the first binomial by each term in the second binomial. This is also known as the FOIL method, which stands for First, Outer, Inner, and Last.
### Step 2: Multiply the Terms
Let's break down the multiplication:
1. First: Multiply the first terms in each binomial:
[tex]\[
(2x) \cdot (7) = 14x
\][/tex]
2. Outer: Multiply the outer terms:
[tex]\[
(2x) \cdot (-4x) = -8x^2
\][/tex]
3. Inner: Multiply the inner terms:
[tex]\[
(5) \cdot (7) = 35
\][/tex]
4. Last: Multiply the last terms in each binomial:
[tex]\[
(5) \cdot (-4x) = -20x
\][/tex]
### Step 3: Combine Like Terms
Now, add all these products together:
[tex]\[
-8x^2 + 14x - 20x + 35
\][/tex]
### Step 4: Simplify the Expression
Combine the [tex]\(x\)[/tex]-terms:
[tex]\[
-8x^2 + (14x - 20x) + 35
\][/tex]
[tex]\[
-8x^2 - 6x + 35
\][/tex]
### Conclusion
The quadratic expression that represents the product of [tex]\((2x + 5)(7 - 4x)\)[/tex] is:
[tex]\[
-8x^2 - 6x + 35
\][/tex]
Given this, the correct answer is:
C. [tex]\(-8x^2 - 6x + 35\)[/tex]
### Step 1: Apply the Distributive Property
We need to multiply each term in the first binomial by each term in the second binomial. This is also known as the FOIL method, which stands for First, Outer, Inner, and Last.
### Step 2: Multiply the Terms
Let's break down the multiplication:
1. First: Multiply the first terms in each binomial:
[tex]\[
(2x) \cdot (7) = 14x
\][/tex]
2. Outer: Multiply the outer terms:
[tex]\[
(2x) \cdot (-4x) = -8x^2
\][/tex]
3. Inner: Multiply the inner terms:
[tex]\[
(5) \cdot (7) = 35
\][/tex]
4. Last: Multiply the last terms in each binomial:
[tex]\[
(5) \cdot (-4x) = -20x
\][/tex]
### Step 3: Combine Like Terms
Now, add all these products together:
[tex]\[
-8x^2 + 14x - 20x + 35
\][/tex]
### Step 4: Simplify the Expression
Combine the [tex]\(x\)[/tex]-terms:
[tex]\[
-8x^2 + (14x - 20x) + 35
\][/tex]
[tex]\[
-8x^2 - 6x + 35
\][/tex]
### Conclusion
The quadratic expression that represents the product of [tex]\((2x + 5)(7 - 4x)\)[/tex] is:
[tex]\[
-8x^2 - 6x + 35
\][/tex]
Given this, the correct answer is:
C. [tex]\(-8x^2 - 6x + 35\)[/tex]
