The given expression for the volume of the storage box is:

$V = l \cdot w \cdot (w + 2)$

We are asked to find the volume when:

$l = 3 \, \text{feet}, \quad w = 2 \, \text{feet}.$

Substituting these values into the formula:

$V = 3 \cdot 2 \cdot (2 + 2) = 3 \cdot 2 \cdot 4.$

Now, calculate the result:

$V = 24 \, \text{cubic feet}.$

The volume of the box is 24 cubic feet.

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How does changing the length $l$ affect the volume?
2. If the width $w$ increases by 1 foot, how would the volume change?
3. What is the general formula for the volume of a box with different side lengths?
4. How does the formula for volume compare to the formula for surface area?
5. Can you derive the formula for volume given different relationships between $l$, $w$, and height?

Tip: When solving problems with multiple variables, substituting known values into the equation step by step helps ensure accuracy.