Choice A is the correct answer. The area of a square is given by $s^2$. We need to write the quadratic expression as a perfect square.

1. Check if $x^2 + 14x + 49$ is a perfect square.
2. Half of 14 is 7. $7^2 = 49$. It matches.
3. Factor: $(x + 7)^2$.

Thus, the side length $s$ is $x + 7$.

Choice B is incorrect because the middle term is positive, so the factor must be $(x+7)$, not $(x-7)$. Choice C is incorrect because $(x+14)^2 = x^2 + 28x + 196$. Choice D is incorrect because it is not the square root of the constant.