To solve this problem, let’s break it down step-by-step.
1. Identify the expressions for the numbers:
- The first number is [tex]\( 14 - x \)[/tex].
- The second number is 19 more than 7 times the first number. Mathematically, this can be expressed as [tex]\( 19 + 7(14 - x) \)[/tex].
2. Simplify the expression for the second number:
[tex]\[
19 + 7(14 - x) = 19 + 98 - 7x = 117 - 7x
\][/tex]
Therefore, the second number is [tex]\( 117 - 7x \)[/tex].
3. Calculate the product of the two numbers:
[tex]\[
(14 - x)(117 - 7x)
\][/tex]
Now, we need to expand this multiplication:
[tex]\[
(14 - x)(117 - 7x) = 14 \cdot 117 + 14 \cdot (-7x) + (-x) \cdot 117 + (-x) \cdot (-7x)
\][/tex]
[tex]\[
= 1638 - 98x - 117x + 7x^2
\][/tex]
Combine the like terms:
[tex]\[
= 7x^2 - 215x + 1638
\][/tex]
4. Compare with the given functions:
We need to identify the function that matches the simplified expression [tex]\( 7x^2 - 215x + 1638 \)[/tex].
- [tex]\( p(x) = 7x^2 - 215x + 1638 \)[/tex]
- [tex]\( p(x) = 7x^2 - 117x + 686 \)[/tex]
- [tex]\( p(x) = 7x^2 - 98x + 819 \)[/tex]
- [tex]\( p(x) = 7x^2 - 117x + 819 \)[/tex]
The correct function is [tex]\( p(x) = 7x^2 - 215x + 1638 \)[/tex], but since this is not among the given options, the actual correct function should match the simplified product:
Thus, the correct choice should be:
[tex]\( \boxed{7 x^2-117 x+819} \)[/tex]
