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------------------------------------------------ Which of the following is the product of [tex]$(7x + 2)$[/tex] and [tex]$(5x - 11)$[/tex]?

A. [tex]$12x^2 - 10x - 77x - 22$[/tex]

B. [tex]$35x^2 - 67x - 22$[/tex]

C. [tex]$12x^2 - 67x - 22$[/tex]

D. [tex]$35x^2 + 67x + 22$[/tex]

Answer :

To find the product of the expressions [tex]\((7x + 2)\)[/tex] and [tex]\((5x - 11)\)[/tex], we can use the distributive property, which is often remembered as the FOIL method (First, Outer, Inner, Last). Let's break it down step by step:

1. First: Multiply the first terms of each binomial:
[tex]\[
7x \cdot 5x = 35x^2
\][/tex]

2. Outer: Multiply the outer terms:
[tex]\[
7x \cdot (-11) = -77x
\][/tex]

3. Inner: Multiply the inner terms:
[tex]\[
2 \cdot 5x = 10x
\][/tex]

4. Last: Multiply the last terms of each binomial:
[tex]\[
2 \cdot (-11) = -22
\][/tex]

Next, we combine the like terms from these products:
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-77x\)[/tex] and [tex]\(10x\)[/tex]:
[tex]\[
-77x + 10x = -67x
\][/tex]

Finally, construct the polynomial from the combined terms:
- The [tex]\(x^2\)[/tex] term: [tex]\(35x^2\)[/tex]
- The combined [tex]\(x\)[/tex] terms: [tex]\(-67x\)[/tex]
- The constant term: [tex]\(-22\)[/tex]

Thus, the product of [tex]\((7x + 2)\)[/tex] and [tex]\((5x - 11)\)[/tex] is:
[tex]\[
35x^2 - 67x - 22
\][/tex]

So, the correct answer is B: [tex]\(35x^2 - 67x - 22\)[/tex].