To find the product of the polynomials \((x^2y + 2)(x^2 - y)\), we can use the distributive property (also known as the FOIL method for binomials).
1. Distribute \(x^2y\) across \(x^2 - y\):
\[
x^2y \cdot x^2 + x^2y \cdot (-y) = x^4y - x^2y^2
\]
2. Distribute \(2\) across \(x^2 - y\):
\[
2 \cdot x^2 + 2 \cdot (-y) = 2x^2 - 2y
\]
Now, combine all the terms from both distributions:
\[
x^4y - x^2y^2 + 2x^2 - 2y
\]
So, the final result is:
\[
x^4y - x^2y^2 + 2x^2 - 2y
\]
Looking through the provided responses, the one that matches this result is:
\[
\text{x}^4\text{y} - \text{x}^2\text{y}^2 + 2\text{x}^2 - 2\text{y}
\]
Thus, the correct answer is:
\[
\text{x}^4\text{y} - \text{x}^2\text{y}^2 + 2\text{x}^2 - 2\text{y}
\]