To simplify the polynomial expression \((xy - 2)(x^2 + 1)\), we will use the distributive property (also known as the FOIL method for binomials). Here are the steps:
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Distribute each term in the first factor \((xy - 2)\) to each term in the second factor \((x^2 + 1)\).
\[ (xy - 2)(x^2 + 1) = (xy)(x^2) + (xy)(1) - (2)(x^2) - (2)(1) \]
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Simplifying each term:
- \(xy \cdot x^2 = xyx^2 = x^3y\)
- \(xy \cdot 1 = xy\)
- \(-2 \cdot x^2 = -2x^2\)
- \(-2 \cdot 1 = -2\)
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Combine all the simplified terms together:
\[ x^3y + xy - 2x^2 - 2 \]
So, the simplified expression is:
\[ x^3y + xy - 2x^2 - 2 \]
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