Expand the product ${6(x+2)(x+3)}$.
First, we use the distributive property to expand the first two factors:

\begin{align*}
6(x+2)(x+3) &= (6\cdot x + 6 \cdot 2) (x+3)\\
&=(6x+12)(x+3)
\end{align*}We use the distributive property again by adding the product of $6x+12$ and $x$ to the product of $6x+12$ and 3:

\begin{align*}
(6x+12)(x+3) &= (6x+12) \cdot x +(6x+12) \cdot 3\\
&= x(6x+12) + 3(6x+12)
\end{align*}We use the distributive property again and combine like terms:

\begin{align*}
x(6x+12) + 3(6x+12) &= 6x^2 + 12x + 18x+ 36\\
&= \boxed{6x^2 + 30x + 36}
\end{align*}