Objective:
To determine the number of roots of the polynomial F(z) = 4z^3 - 8z^2 - z^2 lying outside the unit circle.

Introduction:
The roots of a polynomial are the values of the variable that make the polynomial equal to zero. In this case, we are given a polynomial of degree 3, and we need to find the number of roots that lie outside the unit circle.

Method:
To determine the number of roots outside the unit circle, we can analyze the location of the roots in the complex plane.

Step 1: Factorize the polynomial
First, let's factorize the given polynomial F(z) = 4z^3 - 8z^2 - z^2.

F(z) = z^2(4z - 8 - 1)
= z^2(4z - 9)

Step 2: Set each factor equal to zero
To find the roots of the polynomial, we need to set each factor equal to zero.

z^2 = 0
This equation has a double root at z = 0.

4z - 9 = 0
Solving this equation, we find z = 9/4.

Step 3: Analyze the location of the roots
Now, let's analyze the location of the roots in the complex plane.

Root z = 0:
Since z = 0 lies at the origin, it lies on the unit circle.

Root z = 9/4:
To determine if z = 9/4 lies outside the unit circle, we need to find its magnitude.

|9/4| = 9/4

Since 9/4 is greater than 1, z = 9/4 lies outside the unit circle.

Conclusion:
From the analysis, we found that there is one root (z = 9/4) lying outside the unit circle. Therefore, the correct answer is option 'B' - 1.