To find the product of the polynomials \( (x^2 y + 2)(x^2 - y) \), we will use the distributive property (also known as the FOIL method in the case of binomials) to multiply the two expressions.
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Distribute \( x^2 y \): \[ x^2 y \cdot x^2 = x^4 y \] \[ x^2 y \cdot (-y) = -x^2 y^2 \]
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Distribute \( 2 \): \[ 2 \cdot x^2 = 2x^2 \] \[ 2 \cdot (-y) = -2y \]
Now, combine all these results together: \[ x^4 y - x^2 y^2 + 2x^2 - 2y \]
Putting it all together, the product of the two polynomials is: \[ \boxed{x^4 y - x^2 y^2 + 2x^2 - 2y} \]