If $n \leq 4$ and $2 \leq m \leq n \leq 5,$ then what is the greatest possible value of $(n - m)(n + m)?$

The constraints are:

• $n \leq 4$
• $2 \leq m \leq n \leq 5$

Combining these gives: $2 \leq m \leq n \leq 4$

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The expression can be simplified:

$(n - m)(n + m) = n^2 - m^2$

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To maximize $n^2 - m^2$:

• Maximum value of $n^2$ requires the largest possible $n$
• Minimum value of $m^2$ requires the smallest possible $m$

From the constraints:

• Largest possible $n = 4$
• Smallest possible $m = 2$

Since $2 \leq 4 \leq 4$, these values satisfy all constraints.

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The maximum value is:

$(n - m)(n + m) = n^2 - m^2$

$= 4^2 - 2^2$

$= 16 - 4$

$= 12$

Therefore, the greatest possible value is $12$.