Let's break down the problem Nancy has been given:
1. First number: The problem states that the first number is [tex]\(14 - x\)[/tex].
2. Second number: The second number is described as being 19 more than 7 times the first number. This can be expressed mathematically as:
[tex]\[
19 + 7 \cdot (14 - x)
\][/tex]
3. To simplify the second number:
[tex]\[
19 + 7(14 - x) = 19 + 98 - 7x = 117 - 7x
\][/tex]
So, the second number is [tex]\(117 - 7x\)[/tex].
4. Product of the two numbers:
We need to find the product of the two numbers [tex]\(14 - x\)[/tex] and [tex]\(117 - 7x\)[/tex]:
[tex]\[
(14 - x)(117 - 7x)
\][/tex]
5. Now, let's expand this product:
[tex]\[
(14 - x)(117 - 7x) = 14 \cdot 117 - 14 \cdot 7x - x \cdot 117 + x \cdot 7x
\][/tex]
Breaking this down:
[tex]\[
14 \cdot 117 = 1638
\][/tex]
[tex]\[
14 \cdot 7x = 98x
\][/tex]
[tex]\[
x \cdot 117 = 117x
\][/tex]
[tex]\[
x \cdot 7x = 7x^2
\][/tex]
Putting it all together:
[tex]\[
(14 - x)(117 - 7x) = 1638 - 98x - 117x + 7x^2
\][/tex]
Combine like terms:
[tex]\[
(14 - x)(117 - 7x) = 7x^2 - 215x + 1638
\][/tex]
6. Therefore, the function that represents the product of the two numbers is:
[tex]\[
p(x) = 7x^2 - 215x + 1638
\][/tex]
Upon reviewing the given multiple-choice options, we see:
[tex]\[
\boxed{p(x) = 7x^2 - 215x + 1638}
\][/tex]
So, the correct answer is:
[tex]\[
p(x)=7 x^2-215 x+1,638
\][/tex]
