For this kind of question, a quick approach is to apply the following extremely important property..

If a point lies ON a graph, then the x- and y-coordinates of that point must satisfy the equation of that graph.

So, for example, we can see that the point (-1, 1) lies ON the given graph.
This means $x = -1$ and $y = 1$ must be a solution to the equation of the graph.

Let's plug $x = -1$ and $y = 1$ into each answer choice to see whether it's a solution to that equation. We get:
(A) $1 = –(-1)^2 + 4(-1) + 4$
Evaluate to get: $1 = -1$. Since $x = -1$ and $y = 1$ is NOT a solution to this equation, we can eliminate this answer choice.

(B) $1 = –((-1)+ 2)^2$
Evaluate to get: $1 = -1$. Since $x = -1$ and $y = 1$ is NOT a solution to this equation, we can eliminate this answer choice.

(C) $1 = ((-1) – 2)^2$
Evaluate to get: $1 = 9$. Since $x = -1$ and $y = 1$ is NOT a solution to this equation, we can eliminate this answer choice.

(D) $1 = (-1)^2 – 4(-1) + 4$
Evaluate to get: $1 = 9$. Since $x = -1$ and $y = 1$ is NOT a solution to this equation, we can eliminate this answer choice.

(E) $1 = ((-1) + 2)^2$
Evaluate to get: $1 = 1$. Since $x = -1$ and $y = 1$ IS a solution to this equation.

(F) $1 = (-1)^2 + 4(-1) + 4$
Evaluate to get: $1 = 1$. Since $x = -1$ and $y = 1$ IS a solution to this equation.

Since $y = (x + 2)^2$ and $y = x^2 + 4x + 4$ are equivalent equations, the correct answer here is E and F.