To simplify the polynomial expression \((xy - 2)(x^2 + 1)\), we will use the distributive property (also known as the FOIL method for binomials).
We can distribute \(xy\) and \(-2\) to both terms in \((x^2 + 1)\):
-
Distributing \(xy\):
- \(xy \cdot x^2 = x^3y\)
- \(xy \cdot 1 = xy\)
-
Distributing \(-2\):
- \(-2 \cdot x^2 = -2x^2\)
- \(-2 \cdot 1 = -2\)
Now we can combine these results: \[ (xy - 2)(x^2 + 1) = x^3y + xy - 2x^2 - 2 \]
Rearranging the terms: \[ x^3y - 2x^2 + xy - 2 \]
Thus, filling in the blanks in the format \(x^3y + x^2 + xy + \text{constant}\):
- The coefficient of \(x^3y\) is \(1\),
- The coefficient of \(x^2\) is \(-2\),
- The coefficient of \(xy\) is \(1\),
- The constant term is \(-2\).
Putting it all together, we have: \[ \boxed{1} \quad \boxed{-2} \quad \boxed{1} \quad \boxed{-2} \]