Answer
$12+13\sqrt{2c}+6c$
Work Step by Step
Given \begin{equation}
(3+\sqrt{2 c})(4+\sqrt{18 c}).
\end{equation} Multiply the binomials as shown below.
\begin{equation}
\begin{aligned}
(3+\sqrt{2 c})(4+\sqrt{18 c})&= 4(3+\sqrt{2 c})+\sqrt{18 c}(3+\sqrt{2 c})\\
& =12+4\sqrt{2c}+3\sqrt{9\cdot 2c}+\sqrt{36c^2}\\
& =12 +4\sqrt{2c}+9\sqrt{2c}+6c\\
& = 12+13\sqrt{2c}+6c.
\end{aligned}
\end{equation} The simplified expression is
\begin{equation}
(3+\sqrt{2 c})(4+\sqrt{18 c})=12+13\sqrt{2c}+6c.
\end{equation}
