jmc

First, we use the distributive property to expand the first two factors:

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

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

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