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Answer :
To find the quadratic expression that represents the product of the factors [tex]\((2x + 5)(7 - 4x)\)[/tex], we will expand the expression step-by-step.
1. Apply the Distributive Property:
We need to multiply each term in the first factor by each term in the second factor. Use the distributive property (also known as FOIL for binomials):
[tex]\[
(2x + 5)(7 - 4x) = 2x \cdot 7 + 2x \cdot (-4x) + 5 \cdot 7 + 5 \cdot (-4x)
\][/tex]
2. Calculate Each Product:
- [tex]\(2x \cdot 7 = 14x\)[/tex]
- [tex]\(2x \cdot (-4x) = -8x^2\)[/tex]
- [tex]\(5 \cdot 7 = 35\)[/tex]
- [tex]\(5 \cdot (-4x) = -20x\)[/tex]
3. Combine Like Terms:
Now, combine all the products:
[tex]\[
14x + (-8x^2) + 35 + (-20x)
\][/tex]
Rearrange the terms in descending order of the powers of [tex]\(x\)[/tex]:
[tex]\[
-8x^2 + (14x - 20x) + 35
\][/tex]
Combine the [tex]\(x\)[/tex] terms:
[tex]\[
14x - 20x = -6x
\][/tex]
4. Write the Final Quadratic Expression:
[tex]\[
-8x^2 - 6x + 35
\][/tex]
Based on the options provided, the expression that matches our expansion is:
D. [tex]\(-8x^2 - 6x + 35\)[/tex]
So, the correct quadratic expression that represents the product is option D.
1. Apply the Distributive Property:
We need to multiply each term in the first factor by each term in the second factor. Use the distributive property (also known as FOIL for binomials):
[tex]\[
(2x + 5)(7 - 4x) = 2x \cdot 7 + 2x \cdot (-4x) + 5 \cdot 7 + 5 \cdot (-4x)
\][/tex]
2. Calculate Each Product:
- [tex]\(2x \cdot 7 = 14x\)[/tex]
- [tex]\(2x \cdot (-4x) = -8x^2\)[/tex]
- [tex]\(5 \cdot 7 = 35\)[/tex]
- [tex]\(5 \cdot (-4x) = -20x\)[/tex]
3. Combine Like Terms:
Now, combine all the products:
[tex]\[
14x + (-8x^2) + 35 + (-20x)
\][/tex]
Rearrange the terms in descending order of the powers of [tex]\(x\)[/tex]:
[tex]\[
-8x^2 + (14x - 20x) + 35
\][/tex]
Combine the [tex]\(x\)[/tex] terms:
[tex]\[
14x - 20x = -6x
\][/tex]
4. Write the Final Quadratic Expression:
[tex]\[
-8x^2 - 6x + 35
\][/tex]
Based on the options provided, the expression that matches our expansion is:
D. [tex]\(-8x^2 - 6x + 35\)[/tex]
So, the correct quadratic expression that represents the product is option D.
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