You're very very close, and everything you've done so far is absolutely correct. I think you're just overthinking the process a bit. As you know, when you complete the square you take a polynomial that looks like ax2 + bx and turn it into one that looks like a(x+b/2)2 + k. The given polynomial has an a of 1, so that makes the process that much easier. Let's temporarily forget about the k term and imagine you were asked to complete the square of x2 + 4x. You'd do exactly the same steps you did here, arriving at x2 + 4x + 4 - 4. Now, we'll group up the parts that are a perfect square, leaving (x2 + 4x + 4) - 4, = (x + 2)2 - 4.
However, the exercise didn't ask you to complete the square, per se. It asked you to find a value of k that makes x^2 + 4x + k a perfect square. The perfect square we found above is (x +2)2. What value of k would you use to make the given expression into that perfect square?