Let's simplify the given expression step by step:

(c + 2)c - (c + 3)(c - 3)

First, distribute c to both terms inside the first set of parentheses:
= c^2 + 2c - (c + 3)(c - 3)

Next, distribute the terms inside the second set of parentheses:
= c^2 + 2c - (c^2 - 3c + 3c - 9)

Now, simplify the expression further by combining like terms:
= c^2 + 2c - c^2 + 3c - 3c + 9

Finally, simplify the expression by combining like terms:
= 2c + 9

Therefore, (c + 2)c - (c + 3)(c - 3) simplifies to 2c + 9.