To simplify the expression \((xy - 2)(x^2 + 1)\), we can use the distributive property (also known as the FOIL method for binomials). Here's how it works:
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Distribute \(xy\) to every term in \((x^2 + 1)\):
- \(xy \cdot x^2 = x^3y\)
- \(xy \cdot 1 = xy\)
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Distribute \(-2\) to every term in \((x^2 + 1)\):
- \(-2 \cdot x^2 = -2x^2\)
- \(-2 \cdot 1 = -2\)
Now, combine all the results:
\[ (xy - 2)(x^2 + 1) = x^3y + xy - 2x^2 - 2 \]
Now, we'll place it in the required format:
The simplified polynomial expression is:
\[ x^3y + (-2)x^2 + xy + (-2) \]
So, filling in the blanks:
- ___x^3y: 1
- ___x^2: -2
- ___xy: 1
- ___: -2
Thus, the final answer is: \[ 1, -2, 1, -2 \]
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