The given induction proof is not correct for various reasons. The problem occurs in the base case, which is missing from the proof. In a mathematical induction proof, the base case is the foundation that the rest of the proof builds upon. Without it, the proof is incomplete. Moreover, the statement P(n): "n^2 7n 3 is an even integer" is not true for all n N. For instance, if we substitute n=1, we get 1^2 7*1 3 = 11, which is an odd number, not even. The proof also assumes that the sum of two even numbers (k^2 7k 3 and 2(k 4)) is always even, which is not correct. While it's true that the sum of two even numbers is always even, the problem is that k^2 7k 3 is not necessarily even for all k N, as shown above. So, the induction step is flawed because it's based on an assumption that is not correct. The proof also fails to show that if P(k) is true, then P(k 1) is also true for all k N. In conclusion, the proof is not correct due to the lack of a base case and assumptions in the