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

Answer :

Let's find the product of the factors [tex]\((2x + 5)(7 - 4x)\)[/tex] step by step using the distributive property (also known as the FOIL method for binomials):

1. First Terms: Multiply the first term in each binomial.
[tex]\[
2x \times 7 = 14x
\][/tex]

2. Outer Terms: Multiply the outer terms in the expression.
[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 term in each binomial.
[tex]\[
5 \times (-4x) = -20x
\][/tex]

Next, we combine all these results:

- The quadratic term is [tex]\(-8x^2\)[/tex].
- The linear terms are [tex]\(14x\)[/tex] and [tex]\(-20x\)[/tex]. Combining these, we get:
[tex]\[
14x - 20x = -6x
\][/tex]
- The constant term is [tex]\(35\)[/tex].

Putting it all together, the quadratic expression is:
[tex]\[
-8x^2 - 6x + 35
\][/tex]

Therefore, the quadratic expression that represents the product of the factors [tex]\((2x + 5)(7 - 4x)\)[/tex] is:

C. [tex]\(-8x^2 - 6x + 35\)[/tex]

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