Choice A is the correct answer. We want $a^2 + b^2$.

1. Sum ($a+b$) = 4.
2. Product ($ab$) = 2.
3. Identity: $a^2 + b^2 = (a+b)^2 - 2ab$.
4. Substitute: $4^2 - 2(2) = 16 - 4 = 12$.

Choice B is incorrect because it is $(a+b)^2$. Choice C is incorrect. Choice D is incorrect.