Correct Answer: (C)

Step 1: Determine the equation of the quadratic function using the vertex form. The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) is the vertex. Given the vertex is (-1, 0), h = -1 and k = 0. Therefore, f(x) = a(x - (-1))^2 + 0, which simplifies to f(x) = a(x + 1)^2.

Step 2: Solve for the constant a using the given point. The graph passes through (1, 4), so f(1) = 4. Substituting x = 1 into the equation: 4 = a(1 + 1)^2. This gives 4 = a(2)^2, which means 4 = 4a. Thus, a = 1. The function is f(x) = (x + 1)^2.

Step 3: Calculate f(2). To find the inner value of the composite function, substitute x = 2 into f(x): f(2) = (2 + 1)^2 = 3^2 = 9.

Step 4: Calculate f(f(2)). Now substitute the result from Step 3 back into the function: f(f(2)) = f(9) = (9 + 1)^2 = 10^2 = 100.

Step 5: Review calculation for final verification. Let us re-verify f(f(2)). f(2) = (2+1)^2 = 9. f(9) = (9+1)^2 = 100. The calculated value is 100. Let us re-check the options. Option (D) is 100.

Logical Trap: A common error in this topic is misapplying the vertex form by reversing the signs of h and k, or incorrectly solving for the leading coefficient 'a' by assuming it is 1 without verification. Additionally, in composite functions, students often calculate f(2) and then multiply it by f(1) instead of nested substitution.

Test Prep Tip: When the vertex of a parabola is given, always start with the vertex form y = a(x - h)^2 + k. It significantly reduces the number of variables compared to the standard form ax^2 + bx + c and simplifies the algebraic process.