Answer :
To determine which quadratic expression represents the product of the factors [tex]\((2x + 5)(7 - 4x)\)[/tex], we'll use the distributive property, often referred to as the FOIL method (First, Outer, Inner, Last).
Here are the steps:
1. First terms: Multiply the first terms in each binomial.
- [tex]\(2x \times 7 = 14x\)[/tex]
2. Outer terms: Multiply the outer terms in the binomials.
- [tex]\(2x \times (-4x) = -8x^2\)[/tex]
3. Inner terms: Multiply the inner terms.
- [tex]\(5 \times 7 = 35\)[/tex]
4. Last terms: Multiply the last terms in each binomial.
- [tex]\(5 \times (-4x) = -20x\)[/tex]
Now, add all these products together to form the quadratic expression:
[tex]\[
-8x^2 + 14x - 20x + 35
\][/tex]
Next, combine the like terms [tex]\(14x\)[/tex] and [tex]\(-20x\)[/tex]:
[tex]\[
-8x^2 - 6x + 35
\][/tex]
The quadratic expression that represents the product of these factors is [tex]\(-8x^2 - 6x + 35\)[/tex].
Looking at the provided options, the correct choice is:
D. [tex]\(-8x^2 - 6x + 35\)[/tex]
Here are the steps:
1. First terms: Multiply the first terms in each binomial.
- [tex]\(2x \times 7 = 14x\)[/tex]
2. Outer terms: Multiply the outer terms in the binomials.
- [tex]\(2x \times (-4x) = -8x^2\)[/tex]
3. Inner terms: Multiply the inner terms.
- [tex]\(5 \times 7 = 35\)[/tex]
4. Last terms: Multiply the last terms in each binomial.
- [tex]\(5 \times (-4x) = -20x\)[/tex]
Now, add all these products together to form the quadratic expression:
[tex]\[
-8x^2 + 14x - 20x + 35
\][/tex]
Next, combine the like terms [tex]\(14x\)[/tex] and [tex]\(-20x\)[/tex]:
[tex]\[
-8x^2 - 6x + 35
\][/tex]
The quadratic expression that represents the product of these factors is [tex]\(-8x^2 - 6x + 35\)[/tex].
Looking at the provided options, the correct choice is:
D. [tex]\(-8x^2 - 6x + 35\)[/tex]
