The sum of two numbers is 40 and their difference is 12. What is their product?
Let's start by rewriting this problem into equation form:

\begin{align*}
x + y &= 40, \\
x - y &= 12.
\end{align*}We want to find $xy$, so let's find $x$ and $y$ separately.

Begin by adding the two equations:  \begin{align*}
2x &= 52 \\
x &= 26
\end{align*}Now, subtract the two equations \begin{align*}
2y &= 28 \\
y &= 14
\end{align*}So then $x \cdot y = 26 \cdot 14 = \boxed{364}$.