Given the expression:

[tex]\[
(a + 2b + 3c)^2
\][/tex]

Expand and simplify it.

[tex]\[
\begin{aligned}
(a + 2b + 3c)^2 &= a^2 + 2 \cdot a \cdot 2b + 2 \cdot a \cdot 3c + (2b)^2 + 2 \cdot 2b \cdot 3c + (3c)^2 \\
&= a^2 + 4ab + 6ac + 4b^2 + 12bc + 9c^2 \\
&= a^2 + 4b^2 + 9c^2 + 4ab + 12bc + 6ac
\end{aligned}
\][/tex]

Therefore, the expanded form of [tex]\((a + 2b + 3c)^2\)[/tex] is:

[tex]\[
a^2 + 4b^2 + 9c^2 + 4ab + 12bc + 6ac
\][/tex]