Question

The volume of a rectangular prism is b^3 + 8b^2 + 19b + 12 cubic units, and its height is b + 3 units.

Answer 1

The answer is (b² + 5b + 4).

The volume of a rectangular prism (V) is:
V = l · w · h              (l - length, w - width, h - height)

The base of the rectangular prism is the product of length and width, so the area of the base is:
A = l · w

Since V = l · w · h and A = l · w, then:
V = A · h
It is given:
V = b³ + 8b² + 19b + 12
h = b + 3
⇒ b³ + 8b² + 19b + 12 = A · (b + 3)
⇒ A = (b³ + 8b² + 19b + 12) ÷ (b + 3)

Now, we have to present the volume as multiplication of factors. One of the factors is b+3. So:
b³ + 8b² + 19b + 12 = (b · b² + 3b²) + (5b² + 15b) + (4b + 3·4) =
                                =    b²(b + 3)    +   5b(b + 3)   +  4(b + 3) =
                                = (b + 3)(b² + 5b + 4)

A = (b³ + 8b² + 19b + 12) ÷ (b + 3) = (b + 3)(b² + 5b + 4) ÷ (b + 3)

(b + 3) can be cancelled out:
A = (b² + 5b + 4)