The problem asks to expand the product of the binomial $(2 + 3w)$ and the trinomial $(w^2 + 6w + 9)$ and express the result as a polynomial in standard form.

1. Problem Description

The problem asks to expand the product of the binomial $(2 + 3w)$ and the trinomial $(w^2 + 6w + 9)$ and express the result as a polynomial in standard form.

2. Solution Steps

To expand the expression $(2 + 3w)(w^2 + 6w + 9)$, we use the distributive property (also known as FOIL method when multiplying two binomials). We multiply each term in the first factor $(2 + 3w)$ by each term in the second factor $(w^2 + 6w + 9)$.

$(2 + 3w)(w^2 + 6w + 9) = 2(w^2 + 6w + 9) + 3w(w^2 + 6w + 9)$

$= 2w^2 + 12w + 18 + 3w^3 + 18w^2 + 27w$

Now, we combine like terms:

$= 3w^3 + (2w^2 + 18w^2) + (12w + 27w) + 18$

$= 3w^3 + 20w^2 + 39w + 18$

The polynomial in standard form is $3w^3 + 20w^2 + 39w + 18$.

3. Final Answer

$3w^3 + 20w^2 + 39w + 18$