To simplify the polynomial expression \((xy - 2)(x^2 + 1)\), we can use the distributive property (also known as the FOIL method for binomials).
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Distribute \(xy\) to both terms in the second polynomial: \[ xy \cdot x^2 + xy \cdot 1 = xyx^2 + xy = xy^3 + xy \]
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Distribute \(-2\) to both terms in the second polynomial: \[ -2 \cdot x^2 - 2 \cdot 1 = -2x^2 - 2 \]
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Now, combine all the terms together: \[ xy^3 + xy - 2x^2 - 2 \]
Thus, the simplified expression is: \[ xy^3 + xy - 2x^2 - 2 \]