To simplify the expression \((xy - 2)(x^2 + 1)\), we can use the distributive property (also known as the FOIL method for binomials).
Let's distribute \(xy\) and \(-2\) across \(x^2 + 1\):
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First, multiply \(xy\) by each term in \(x^2 + 1\): \[ xy \cdot x^2 = xyx^2 = x^3y \] \[ xy \cdot 1 = xy \]
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Next, multiply \(-2\) by each term in \(x^2 + 1\): \[ -2 \cdot x^2 = -2x^2 \] \[ -2 \cdot 1 = -2 \]
Now combine all the terms together: \[ x^3y + xy - 2x^2 - 2 \]
Arranging it in standard form gives: \[ x^3y - 2x^2 + xy - 2 \]
Thus, the simplified polynomial expression is: \[ x^3y - 2x^2 + xy - 2 \]