To simplify the polynomial expression \((xy - 2)(x^2 + 1)\), we can distribute each term in the first polynomial to 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, we simplify each term:
- \(xy \cdot x^2 = x^3y\)
- \(xy \cdot 1 = xy\)
- \(-2 \cdot x^2 = -2x^2\)
- \(-2 \cdot 1 = -2\)
Now, combine all the terms:
\[ x^3y + xy - 2x^2 - 2 \]
Now, let's write it in the specified format:
\[ \boxed{1}x^3y + \boxed{-2}x^2 + \boxed{1}xy + \boxed{-2} \]
So, filling the boxes with the coefficients from the simplified expression, we have:
- For \(x^3y\), the coefficient is 1.
- For \(x^2\), the coefficient is -2.
- For \(xy\), the coefficient is 1.
- For the constant term, the coefficient is -2.
Thus, the filled boxes are:
\[ 1, -2, 1, -2 \]