To simplify the polynomial expression \((xy - 2)(x^2 + 1)\), we can use the distributive property (also known as the FOIL method for binomials). We will distribute each term in the first polynomial by each term in the second polynomial.
\[ (xy - 2)(x^2 + 1) = xy \cdot x^2 + xy \cdot 1 - 2 \cdot x^2 - 2 \cdot 1 \]
Now, perform the multiplication:
- \(xy \cdot x^2 = xyx^2 = x^3y\)
- \(xy \cdot 1 = xy\)
- \(-2 \cdot x^2 = -2x^2\)
- \(-2 \cdot 1 = -2\)
Now, combine all the results together:
\[ x^3y + xy - 2x^2 - 2 \]
Thus, the simplified expression is:
\[ x^3y + xy - 2x^2 - 2 \]