College

Which quadratic expression represents the product of these expressions [tex]\((2x + 5)(7 - 4x)\)[/tex]?

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

Answer :

To solve the problem of expanding the expression [tex]\((2x + 5)(7 - 4x)\)[/tex] and finding which option represents the product, we'll use the distributive property (also known as the FOIL method for binomials). Let's go through the steps:

1. Multiply each term in the first binomial by each term in the second binomial.

- Multiply the first term of the first binomial by the first term of the second binomial:
[tex]\[
2x \times 7 = 14x
\][/tex]

- Multiply the first term of the first binomial by the second term of the second binomial:
[tex]\[
2x \times (-4x) = -8x^2
\][/tex]

- Multiply the second term of the first binomial by the first term of the second binomial:
[tex]\[
5 \times 7 = 35
\][/tex]

- Multiply the second term of the first binomial by the second term of the second binomial:
[tex]\[
5 \times (-4x) = -20x
\][/tex]

2. Combine all the products.

The expression becomes:
[tex]\[
14x + (-8x^2) + 35 + (-20x)
\][/tex]

3. Combine like terms and arrange in standard quadratic form.

- Combine the [tex]\(x\)[/tex]-terms: [tex]\(14x - 20x = -6x\)[/tex]

- The expression in standard quadratic form is:
[tex]\[
-8x^2 - 6x + 35
\][/tex]

So, the quadratic expression that represents the product is [tex]\(-8x^2 - 6x + 35\)[/tex].

Let's identify which option corresponds to this expression:

- Option A: [tex]\(-8x^2 + 6x - 35\)[/tex]
- Option B: [tex]\(-8x^2 - 6x + 35\)[/tex]
- Option C: [tex]\(-8x^2 + 34x - 35\)[/tex]
- Option D: [tex]\(-8x^2 - 34x + 35\)[/tex]

The correct option is B: [tex]\(-8x^2 - 6x + 35\)[/tex].