Let the two numbers be $a$ and $b$.

Given:

• $a + b = 22$
• $a^2 + b^2 = 260$

Use the identity $(a + b)^2 = a^2 + 2ab + b^2$:

$22^2 = 260 + 2ab$

$484 = 260 + 2ab$

$2ab = 224$

$ab = 112$

The product of the two numbers is $112$.