High School

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------------------------------------------------ Select the correct answer.

Which quadratic expression represents the product of these factors?

[tex](2x + 5)(7 - 4x)[/tex]

A. [tex]-8x^2 - 6x + 35[/tex]
B. [tex]-8x^2 + 34x - 35[/tex]
C. [tex]-8x^2 - 34x + 35[/tex]
D. [tex]-8x^2 + 6x - 35[/tex]

Answer :

To find the product of the factors

[tex]$$ (2x+5)(7-4x), $$[/tex]

we apply the distributive property (also known as the FOIL method). Here are the steps:

1. Multiply the first term of the first factor by each term of the second factor:
- [tex]$$ 2x \cdot 7 = 14x $$[/tex]
- [tex]$$ 2x \cdot (-4x) = -8x^2 $$[/tex]

2. Multiply the second term of the first factor by each term of the second factor:
- [tex]$$ 5 \cdot 7 = 35 $$[/tex]
- [tex]$$ 5 \cdot (-4x) = -20x $$[/tex]

3. Combine the like terms (the [tex]$x$[/tex] terms):
- From step 1 and step 2, the [tex]$x$[/tex] terms are: [tex]$$ 14x \text{ and } -20x $$[/tex]
- Adding them gives: [tex]$$ 14x - 20x = -6x $$[/tex]

4. Write the final expanded quadratic expression:
- The [tex]$x^2$[/tex] term is: [tex]$$ -8x^2 $$[/tex]
- The [tex]$x$[/tex] term is: [tex]$$ -6x $$[/tex]
- The constant term is: [tex]$$ 35 $$[/tex]

Thus, the product is:

[tex]$$ -8x^2 - 6x + 35. $$[/tex]

Comparing with the given options, we see that option A is:

[tex]$$ -8x^2 - 6x + 35. $$[/tex]

So, the correct answer is option A.