To solve the given problem, let's analyze the provided equations:

• The first equation is \(x^4 + x^2y^2 + y^4 = 8\).
• The second equation is \(x^2 + xy + y^2 = 4\).

We need to find the value of \(xy\).

Let's solve this step-by-step:

1. Consider the identity: \((x^2 + y^2)^2 = x^4 + 2x^2y^2 + y^4\).
2. Let's express \(x^4 + x^2y^2 + y^4\) using this identity:
   • We can rewrite it as: \(x^4 + x^2y^2 + y^4 = (x^2 + y^2)^2 - x^2y^2\).
3. From the second equation \(x^2 + xy + y^2 = 4\), we know \(x^2 + y^2\) can be expressed in terms of \(xy\):
   • Squaring both sides of \(x^2 + xy + y^2 = 4\), we get:
   • \((x^2 + xy + y^2)^2 = 16\)
   • Expand to find:
     \(x^4 + 2x^3y + 3x^2y^2 + 2xy^3 + y^4 = 16\)
      
4. Now, notice:
   \(x^4 + x^2y^2 + y^4 = (x^2 + y^2)^2 - x^2y^2 = 8\)
   \((x^2 + y^2)^2 = 8 + x^2y^2\) (from the given equation).
5. Let \(s = x^2 + y^2\). Then \(s = 4 - xy\). From the identity:
   \((4 - xy)^2 = 8 + x^2y^2\) (substituting).
6. Let's solve for \(xy\):
   • \(16 - 8xy + (xy)^2 = 8 + (xy)^2\)
   • Cancel \((xy)^2\) from both sides, then \(16 - 8xy = 8\).
   • Simplify to find \(8 = 8xy\).
   • Finally, \(xy = 1\).

Thus, the value of \(xy\) is 1. The correct answer is option: 1.