Question

The volume of a rectangular prism is b3 + 8b2 + 19b + 12 cubic units, and its height is b + 3 units. The area of the base of the rectangular prism is square units.

Answer 1

Given:

The volume of the rectangular prism is 

$b^{3}+8 b^{2} +19b+12$,

the height is h=(b+3)

1. The volume of a rectangular prism is (base area)*height

also, notice that the volume is a third degree polynomial, the height is a 1st degree polynomial, so the base area must be a 2nd degree polynomial, whose coefficients we don't know yet.
Let this quadratic polynomial be $(mb^{2}+nb+k)$

 
2

$b^{3}+8 b^{2} +19b+12=(mb^{2}+nb+k)*(b+3)$


notice that  $b^{3}$ is the product of the largest 2 terms: $mb^{2}$ and b, so m must be 1

also, notice that 12 is the product of the constants, k and 3

so k*3=12, this means k=4

3
we write the above equality again:

$b^{3}+8 b^{2} +19b+12=(b^{2}+nb+4)*(b+3)$


$(b^{2}+nb+4)(b+3)= b^{3}+3 b^{2} +nb^{2}+3nb+4b+12$

=$= b^{3}+(n+3)b^{2}+(3n+4)b+12$


4
now compare the coefficient with the left side:

$8 b^{2}=(n+3)b^{2}$

8=n+3

n=5


substituting n=5: 

the base area is $b^{2}+5b+4$

Answer: $b^{2}+5b+4$