Factorize the Volume Expression: b³ + 5b² + 6b = v cm³

Let's address the given box volume expression:

$V=b^3+5b^2+6b$

Proceed to break it down into factors. Note that we can factor out a common term from all members in the volume expression.

It is the common factor b that is the greatest for both numbers and letters,

$V=b^3+5b^2+6b \\ \downarrow\\ \boxed{ V=b(b^2+5b+6)}$

Let's continue to address the expression inside of the parentheses:

$b^2+5b+6$

Note that the coefficient of the squared term in this expression is 1, therefore we can (try to) factor this expression by using quick trinomial factoring:

We'll look for a pair of numbers whose product is the free term in the expression on the left side, and whose sum is the coefficient of the first-degree term in the expression meaning two numbers $m,\hspace{2pt}n$ that satisfy the following values:

$m\cdot n=6\\ m+n=5$

From the first requirement above, meaning - from the multiplication, we can deduce according to the rules of sign multiplication that both numbers have the same signs. Now we'll recall that 6 has the factors (whole numbers) 2 and 3 or 6 and 1, fulfilling the second requirement mentioned. Together with the fact that the signs of the numbers we're looking for are equal to each other leads us to the conclusion that the only possibility for the two numbers we're looking for is:

$\begin{cases} m=2\\ n=3 \end{cases}$

Therefore we'll factor the expression in question to:

$b^2+5b+6 \\ \downarrow\\ (b+2)(b+3)$

where we used the pair of numbers we found earlier in this factoring,

Let's return now to the volume expression we started to factor earlier (highlighted with a square) and apply this factoring:

$$$V=b(b^2+5b+6)\\ \downarrow\\ \boxed{V=b(b+2)(b+3)}$

Note that this is indeed the most factored expression possible for the given volume expression,

Therefore the correct answer is answer D