To find the product of the polynomials \((x^2y + 2)(x^2 - y)\), we will apply the distributive property (also known as the FOIL method for binomials).
Let's distribute each term in the first polynomial to each term in the second polynomial:
1. **Multiply \(x^2y\) by \(x^2\)**:
\[
x^2y \cdot x^2 = x^4y
\]
2. **Multiply \(x^2y\) by \(-y\)**:
\[
x^2y \cdot (-y) = -x^2y^2
\]
3. **Multiply \(2\) by \(x^2\)**:
\[
2 \cdot x^2 = 2x^2
\]
4. **Multiply \(2\) by \(-y\)**:
\[
2 \cdot (-y) = -2y
\]
Now combine all these results:
\[
x^4y - x^2y^2 + 2x^2 - 2y
\]
Thus, the final expression after combining all terms is:
\[
x^4y - x^2y^2 + 2x^2 - 2y
\]
So the product of the polynomials \((x^2y + 2)(x^2 - y)\) is:
\[
\boxed{x^4y - x^2y^2 + 2x^2 - 2y}
\]