For the equation 4x² + 16x + 4m = 0 or x² + 4x + m = 0 to have real roots, b² – 4ac must be greater than or equal to 0, i.e. 16 - 4m ≥ 0 or 4 ≥ m.

Given:
- Quadratic equation: 4x^2 + 16x + 4m = 0
- m is a non-negative integer
- We need to find the number of possible values for m such that the quadratic equation has real roots.

Analysis:
For a quadratic equation to have real roots, the discriminant (b^2 - 4ac) must be greater than or equal to zero. In this case, the discriminant is:

Discriminant = (16^2) - (4 * 4 * m) = 256 - 16m

So, we need to find the values of m for which the discriminant is greater than or equal to zero.

Solution:
To find the number of possible values for m, we will analyze the discriminant in different ranges.

Case 1: Discriminant > 0
When the discriminant is greater than zero, the quadratic equation has two distinct real roots.
256 - 16m > 0
Simplifying, we get:
m < />

Case 2: Discriminant = 0
When the discriminant is equal to zero, the quadratic equation has two equal real roots.
256 - 16m = 0
Simplifying, we get:
m = 16

Case 3: Discriminant < />
When the discriminant is less than zero, the quadratic equation has complex roots and no real roots.
256 - 16m < />
Simplifying, we get:
m > 16

Summary:
From the analysis, we can conclude that:
- For m < 16,="" the="" discriminant="" is="" greater="" than="" zero,="" and="" the="" quadratic="" equation="" has="" two="" distinct="" real="" />
- For m = 16, the discriminant is equal to zero, and the quadratic equation has two equal real roots.
- For m > 16, the discriminant is less than zero, and the quadratic equation has no real roots.

Number of Possible Values for m:
From the above analysis, we can see that there are 16 possible values for m (0 to 15) in which the quadratic equation has real roots (two distinct or two equal). Therefore, the correct answer is option B - 5.