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To find the product of the polynomials \((x^2y + 2)(x^2 - y)\), we will use the distributive property (also known as the FOIL method for binomials).
Let's compute:
\[
(x^2y + 2)(x^2 - y) = x^2y \cdot x^2 + x^2y \cdot (-y) + 2 \cdot x^2 + 2 \cdot (-y)
\]
Now, let's calculate each term:
1. \(x^2y \cdot x^2 = x^{4}y\)
2. \(x^2y \cdot (-y) = -x^2y^2\)
3. \(2 \cdot x^2 = 2x^2\)
4. \(2 \cdot (-y) = -2y\)
Now, combining all these terms together:
\[
x^{4}y - x^{2}y^{2} + 2x^{2} - 2y
\]
Thus, the product of the polynomials \((x^2y + 2)(x^2 - y)\) is:
\[
\boxed{x^{4}y - x^{2}y^{2} + 2x^{2} - 2y}
\]