College

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 :

To find the quadratic expression representing the product of the factors [tex]\((2x + 5)(7 - 4x)\)[/tex], we will expand the expression using the distributive property, often referred to as the FOIL method. FOIL stands for First, Outside, Inside, Last, which are the elements you need to multiply together. Here's how to do it step by step:

1. First: Multiply the first terms in each set of parentheses:
[tex]\[
2x \times 7 = 14x
\][/tex]

2. Outside: Multiply the outer terms in the multiplication:
[tex]\[
2x \times (-4x) = -8x^2
\][/tex]

3. Inside: Multiply the inner terms:
[tex]\[
5 \times 7 = 35
\][/tex]

4. Last: Multiply the last terms in each set of parentheses:
[tex]\[
5 \times (-4x) = -20x
\][/tex]

Now, combine all these results into a single quadratic expression:

[tex]\[
-8x^2 + 14x + 35 - 20x
\][/tex]

Next, combine like terms (the terms involving [tex]\(x\)[/tex]):

[tex]\[
14x - 20x = -6x
\][/tex]

So, the quadratic expression simplifies to:

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

Now, we can compare this result to the given options:

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]

The correct quadratic expression, [tex]\(-8x^2 - 6x + 35\)[/tex], matches option C. So, the answer is:

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

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