Relationship Between Quadratic Function Outputs

Let function $f(x) = ax^2 + bx + c$ represent a quadratic function, where $a$, $b$, and $c$ are real numbers. We are given the conditions:

1. $f(1) = 0$

2. $f(-1) = 0$

3. $b = -2a$

Using these conditions, determine the relationship between the following two quantities:

Quantity A: $f(2)$

Quantity B: $4$

Based on the value of $a$, determine the relationship between Quantity A and Quantity B.