To multiply the polynomials \((x^2y + 2)(x^2 - y)\), we can use the distributive property (also known as the FOIL method for binomials).
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Distribute \(x^2y\) across \(x^2 - y\):
- \(x^2y \cdot x^2 = x^4y\)
- \(x^2y \cdot (-y) = -x^2y^2\)
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Distribute \(2\) across \(x^2 - y\):
- \(2 \cdot x^2 = 2x^2\)
- \(2 \cdot (-y) = -2y\)
Next, we combine all these results: \[ x^4y - x^2y^2 + 2x^2 - 2y \]
Thus, the result is: \[ x^4y - x^2y^2 + 2x^2 - 2y \]
Now, let's match it to your provided options:
- \(x^4y - xy + x^2y\) (does not match)
- \(x^3y - x^2y^2 + 2x^2 - 2y\) (does not match, while \(x^3y\) term is incorrect)
- \(x^4y + x^2 - 2y\) (does not match, missing \( -x^2y^2\))
- \(x^4y - x^2y^2 + 2x^2 - 2y\) (this one matches with the derived result)
Therefore, the correct simplified expression is: \[ x^4y - x^2y^2 + 2x^2 - 2y \]
So, the final answer is: \[ \boxed{x^4y - x^2y^2 + 2x^2 - 2y} \]