To simplify the polynomial expression \((xy - 2)(x^2 + 1)\), we can use the distributive property (also known as the FOIL method for binomials):
\[ (xy - 2)(x^2 + 1) = xy \cdot x^2 + xy \cdot 1 - 2 \cdot x^2 - 2 \cdot 1 \]
Now, we can calculate each term:
- \(xy \cdot x^2 = x^3y\)
- \(xy \cdot 1 = xy\)
- \(-2 \cdot x^2 = -2x^2\)
- \(-2 \cdot 1 = -2\)
Putting it all together, we get:
\[ x^3y + xy - 2x^2 - 2 \]
So, the simplified form of the polynomial expression \((xy - 2)(x^2 + 1)\) is:
\[ x^3y - 2x^2 + xy - 2 \]
Thus, if we write the final result with coefficients as you requested, we can represent it like this:
\[ x^3y + xy - 2x^2 - 2 \]
So the simplified expression does not require more than what was computed; it is already presented in a simplified manner.
In summary, the simplified expression is:
\[ x^3y + xy - 2x^2 - 2 \]