To find the range of the function f(x) = sin^{-1}x * 2tan^{-1}x * x^2 * 4x * 1, we need to consider the range of each individual term and then find the combined range.

1. Range of sin^{-1}x:
The range of the arcsine function is [-pi/2, pi/2]. This means that the function sin^{-1}x will give values between -pi/2 and pi/2 for all values of x.

2. Range of 2tan^{-1}x:
The range of the arctangent function is (-pi/2, pi/2). When multiplied by 2, the range becomes (-pi, pi). This means that the function 2tan^{-1}x will give values between -pi and pi for all values of x.

3. Range of x^2:
The range of x^2 is [0, infinity). This means that the function x^2 will give values greater than or equal to 0 for all values of x.

4. Range of 4x:
The range of 4x is (-infinity, infinity). This means that the function 4x will give all real values for all values of x.

5. Range of 1:
The range of a constant function is the single value that it represents. In this case, the function 1 will always give the value 1.

Now, to find the combined range, we need to consider the intersection of the ranges of the individual terms.

- The intersection of the ranges of sin^{-1}x and 2tan^{-1}x is (-pi/2, pi/2).
- The intersection of this range and the range of x^2 is [0, pi/2].
- The intersection of this range and the range of 4x is [0, pi/2].
- The intersection of this range and the range of 1 is [0, pi/2].

Therefore, the range of the function f(x) is [0, pi/2].

However, the question asks for the value of a + b. Since the range is [0, pi/2], a = 0 and b = pi/2. Therefore, a + b = 0 + pi/2 = pi/2.

Hence, the correct answer is 4.