Question

the dimensions of a rectangle are represented by the functions shown.
rectangle image with f(x) = 3x - 2 (vertical side) and g(x) = 4x + 5 (horizontal side)
which function represents the area of the rectangle?
○ $12x^2 + 15x - 10$
○ $12x^2 + 7x + 10$
○ $12x^2 - 8x - 10$
○ $12x^2 + 7x - 10$

Explanation:

Step1: Recall area of rectangle formula

The area \( A \) of a rectangle is the product of its length and width. Here, the length and width are given by the functions \( f(x) = 3x - 2 \) and \( g(x) = 4x + 5 \), so \( A(x)=f(x)\times g(x)=(3x - 2)(4x + 5) \).

Step2: Expand the product

Using the distributive property (FOIL method):
\[

$$\begin{align*} (3x - 2)(4x + 5)&=3x\times4x+3x\times5-2\times4x-2\times5\\ &=12x^{2}+15x - 8x - 10\\ &=12x^{2}+7x - 10 \end{align*}$$

\]

Answer:

\( 12x^{2}+7x - 10 \) (the last option: \( 12x^{2}+7x - 10 \))