Answer :
Let's begin by multiplying the two given factors: [tex]\((2x + 5)\)[/tex] and [tex]\((7 - 4x)\)[/tex].
To do this, we will use the distributive property (also known as the FOIL method in the context of binomials). Here are the steps:
1. First, multiply the first terms from each binomial:
[tex]\[
2x \cdot 7 = 14x
\][/tex]
2. Outer, multiply the outer terms from each binomial:
[tex]\[
2x \cdot (-4x) = -8x^2
\][/tex]
3. Inner, multiply the inner terms from each binomial:
[tex]\[
5 \cdot 7 = 35
\][/tex]
4. Last, multiply the last terms from each binomial:
[tex]\[
5 \cdot (-4x) = -20x
\][/tex]
Now, add all these products together to form the quadratic expression:
[tex]\[
14x - 8x^2 + 35 - 20x
\][/tex]
Next, combine like terms:
[tex]\[
-8x^2 + 14x - 20x + 35
\][/tex]
Simplify the expression:
[tex]\[
-8x^2 - 6x + 35
\][/tex]
So, the quadratic expression representing the product of [tex]\((2x + 5)\)[/tex] and [tex]\((7 - 4x)\)[/tex] is:
[tex]\[
-8x^2 - 6x + 35
\][/tex]
Therefore, the correct answer is:
D. [tex]\(-8x^2 - 6x + 35\)[/tex]
To do this, we will use the distributive property (also known as the FOIL method in the context of binomials). Here are the steps:
1. First, multiply the first terms from each binomial:
[tex]\[
2x \cdot 7 = 14x
\][/tex]
2. Outer, multiply the outer terms from each binomial:
[tex]\[
2x \cdot (-4x) = -8x^2
\][/tex]
3. Inner, multiply the inner terms from each binomial:
[tex]\[
5 \cdot 7 = 35
\][/tex]
4. Last, multiply the last terms from each binomial:
[tex]\[
5 \cdot (-4x) = -20x
\][/tex]
Now, add all these products together to form the quadratic expression:
[tex]\[
14x - 8x^2 + 35 - 20x
\][/tex]
Next, combine like terms:
[tex]\[
-8x^2 + 14x - 20x + 35
\][/tex]
Simplify the expression:
[tex]\[
-8x^2 - 6x + 35
\][/tex]
So, the quadratic expression representing the product of [tex]\((2x + 5)\)[/tex] and [tex]\((7 - 4x)\)[/tex] is:
[tex]\[
-8x^2 - 6x + 35
\][/tex]
Therefore, the correct answer is:
D. [tex]\(-8x^2 - 6x + 35\)[/tex]
