Question

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

Explanation:

Step1: Recall area of rectangle

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

Step2: Multiply the two binomials

We use the distributive property (FOIL method) to multiply \( (3x - 2)(4x + 5) \).
First terms: \( 3x \times 4x = 12x^2 \)
Outer terms: \( 3x \times 5 = 15x \)
Inner terms: \( -2 \times 4x = -8x \)
Last terms: \( -2 \times 5 = -10 \)
Now, combine like terms: \( 12x^2 + 15x - 8x - 10 = 12x^2 + 7x - 10 \)

Answer:

\( 12x^2 + 7x - 10 \) (the fourth option: \( 12x^2 + 7x - 10 \))