To expand the expression \((xy - 2)(x^2 + 1)\), we can use the distributive property (also known as the FOIL method for binomials). Let's break it down:
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Distribute \(xy\) to both terms in the second factor \((x^2 + 1)\): \[ xy \cdot x^2 + xy \cdot 1 = xyx^2 + xy = xy^1x^2 + xy = xy^1x^2 + xy = xyx^2 + xy \]
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Distribute \(-2\) to both terms in the second factor \((x^2 + 1)\): \[ -2 \cdot x^2 + (-2) \cdot 1 = -2x^2 - 2 \]
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Now, combine all the terms obtained from the distribution: \[ xyx^2 + xy - 2x^2 - 2 \]
Therefore, the expanded form of \((xy - 2)(x^2 + 1)\) is: \[ xyx^2 - 2x^2 + xy - 2 \]
This can also be written as: \[ x^3y - 2x^2 + xy - 2 \]
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