Question

select the correct answer. consider these shapes. shape a has side lengths $3x^2 + 2$ and $7x^2 + 4x + 8$. shape b has side length $3x^2 + 2$ (and is a square). if shape b is a square, which polynomial represents the difference of the areas of shape a and shape b? a. $32x^2 + 8x + 28$ b. $12x^4 + 12x^3 + 26x^2 + 8x + 12$ c. $8x^2 + 8x + 12$ d. $30x^4 + 12x^3 + 40x^2 + 8x + 20$

Explanation:

Step1: Calculate Area of Shape A

Area = $(3x^2+2)(7x^2+4x+8)$
$=3x^2(7x^2+4x+8)+2(7x^2+4x+8)$
$=21x^4+12x^3+24x^2+14x^2+8x+16$
$=21x^4+12x^3+38x^2+8x+16$

Step2: Calculate Area of Shape B

Area = $(3x^2+2)^2$
$=(3x^2)^2+2(3x^2)(2)+2^2$
$=9x^4+12x^2+4$

Step3: Find Area Difference (A-B)

Difference = $(21x^4+12x^3+38x^2+8x+16)-(9x^4+12x^2+4)$
$=21x^4-9x^4+12x^3+38x^2-12x^2+8x+16-4$
$=12x^4+12x^3+26x^2+8x+12$

Answer:

B. $12x^4 + 12x^3 + 26x^2 + 8x + 12$