High School

The following expressions are factors of a polynomial. Find the product of [tex]$2x+5$[/tex] and [tex]$x-7$[/tex].

A. [tex]$2x^2 + 5x - 35$[/tex]
B. [tex]$3x - 2$[/tex]
C. [tex]$2x^2 + 19x - 35$[/tex]
D. [tex]$2x^2 - 9x - 35$[/tex]

Answer :

To find the product of [tex]\((2x + 5)\)[/tex] and [tex]\((x + 7)\)[/tex], we need to use the distributive property (also known as the FOIL method for binomials). Let's break it down step-by-step.

1. First, multiply each term in the first binomial by each term in the second binomial:
[tex]\[
(2x + 5)(x + 7)
\][/tex]

2. Distribute [tex]\(2x\)[/tex] to each term in the second binomial:
- [tex]\(2x \cdot x = 2x^2\)[/tex]
- [tex]\(2x \cdot 7 = 14x\)[/tex]

3. Distribute [tex]\(5\)[/tex] to each term in the second binomial:
- [tex]\(5 \cdot x = 5x\)[/tex]
- [tex]\(5 \cdot 7 = 35\)[/tex]

4. Combine all these products:
[tex]\[
2x^2 + 14x + 5x + 35
\][/tex]

5. Combine like terms ([tex]\(14x\)[/tex] and [tex]\(5x\)[/tex]):
[tex]\[
2x^2 + 19x + 35
\][/tex]

So, the product of [tex]\((2x + 5)\)[/tex] and [tex]\((x + 7)\)[/tex] is:
[tex]\[
2x^2 + 19x + 35
\][/tex]

Hence, the correct expression from the given options is:
[tex]\[
2x^2 + 19x - 35
\][/tex]