Question

1. find the area of the picture frame. write your answer in standard form. image of a picture frame with outer dimensions 5x + 3 (length) and 3x + 1 (width), and inner dimensions reduced by x on each side a) 3x² + 6x + 3 square units b) 12x² + 8x square units c) 15x² + 14x + 3 square units d) 7x² + 4x square units

Explanation:

Step1: Find outer rectangle area

Area = length × width
$\text{Outer Area} = (5x+3)(3x+1)$
Expand using FOIL:

$$\begin{align*} (5x+3)(3x+1)&=5x(3x)+5x(1)+3(3x)+3(1)\\ &=15x^2 + 5x + 9x + 3\\ &=15x^2 + 14x + 3 \end{align*}$$

Step2: Find inner rectangle dimensions

Inner length = $5x+3 - 2x = 3x+3$
Inner width = $3x+1 - 2x = x+1$

Step3: Find inner rectangle area

$\text{Inner Area} = (3x+3)(x+1)$
Expand:

$$\begin{align*} (3x+3)(x+1)&=3x(x)+3x(1)+3(x)+3(1)\\ &=3x^2 + 3x + 3x + 3\\ &=3x^2 + 6x + 3 \end{align*}$$

Step4: Calculate frame area (outer - inner)

$\text{Frame Area} = (15x^2 + 14x + 3) - (3x^2 + 6x + 3)$

$$\begin{align*} &=15x^2 + 14x + 3 - 3x^2 - 6x - 3\\ &=12x^2 + 8x \end{align*}$$

Answer:

B. $12x^2 + 8x$ square units