To simplify the polynomial expression \((xy - 2)(x^2 + 1)\), we will use the distributive property (also known as the FOIL method for binomials).
\[ (xy - 2)(x^2 + 1) \]
Distributing each term in the first polynomial to each term in the second polynomial:
- \(xy \cdot x^2 = x^3y\)
- \(xy \cdot 1 = xy\)
- \(-2 \cdot x^2 = -2x^2\)
- \(-2 \cdot 1 = -2\)
Now, we can combine all these results:
\[ x^3y + xy - 2x^2 - 2 \]
Now, we can write it in descending order by the power of \(x\):
\[ x^3y - 2x^2 + xy - 2 \]
Thus, the simplified expression is:
\[ x^3y - 2x^2 + xy - 2 \]
In the form you provided, it can be filled in as:
- Coefficient of \(x^3y\) is \(1\)
- Coefficient of \(x^2\) is \(-2\)
- Coefficient of \(xy\) is \(1\)
- Constant term is \(-2\)
So the final representation is:
\[ [x^3y] + [-2]x^2 + [1]xy + [-2] \]
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