Question

select the correct answer.
rhea is solving a math puzzle. to find the solution of the puzzle, she must find the product of two numbers. the first number is the sum of 23 and x, and the second number is 18 less than two times the first number. which of the following functions represents the product of these two numbers?
a. ( p(x) = 2x^2 + 28x - 414 )
b. ( p(x) = 2x^2 + 102x + 1288 )
c. ( p(x) = 2x^2 + 110x + 1472 )
d. ( p(x) = 2x^2 + 74x + 644 )

Explanation:

Step1: Define the first number

The first number is the sum of 23 and \( x \), so it is \( 23 + x \).

Step2: Define the second number

The second number is 18 less than two times the first number. Two times the first number is \( 2(23 + x) \), so the second number is \( 2(23 + x)- 18 \). Let's simplify this: \( 2(23 + x)-18=46 + 2x-18 = 2x + 28 \).

Step3: Find the product function \( P(x) \)

The product of the two numbers is \( P(x)=(23 + x)(2x + 28) \). We expand this using the distributive property (FOIL method):
\[

$$\begin{align*} (23 + x)(2x + 28)&=23\times(2x)+23\times28+x\times(2x)+x\times28\\ &=46x + 644+2x^{2}+28x\\ &=2x^{2}+(46x + 28x)+644\\ &=2x^{2}+74x + 644 \end{align*}$$

\]

Answer:

D. \( P(x)=2x^{2}+74x + 644 \)