Final answer:
To determine the width of a rectangle with a long side of 2b + 4 and an area of 4b² + 12b + 8, divide the area by the length to get the width, which simplifies to 2b + 2; hence, the correct answer is option C (2b + 2).
Step-by-step explanation:
The long side of a rectangle is given as 2b + 4 units, and the area of the rectangle is 4b² + 12b + 8. To find the width of the rectangle, we divide the area by the length of the long side:
- Area of rectangle = Length × Width
- 4b² + 12b + 8 = (2b + 4) × Width
- Width = ⅔(4b² + 12b + 8) / (2b + 4)
- Width = (4b² + 12b + 8) / (2b + 4)
To simplify, we factor the area polynomial:
- 4b² + 12b + 8 = 4(b² + 3b + 2)
- 4(b² + 3b + 2) = 4(b+1)(b+2)
- Now divide by (2b + 4), which is equivalent to 2(b+2)
- (4(b+1)(b+2)) / (2(b+2))
- (2(b+1)(b+2)) / ((b+2))
- Width = 2(b+1) or 2b + 2 (the (b+2) terms cancel out)
Therefore, the correct answer is option C (2b + 2).