To simplify the polynomial expression \((xy-2)(x^{2}+1)\), we will use the distributive property (also known as the FOIL method for two binomials) to expand the expression.
We start by distributing each term in the first parenthesis with each term in the second parenthesis:
\[ (xy - 2)(x^2 + 1) = xy \cdot x^2 + xy \cdot 1 - 2 \cdot x^2 - 2 \cdot 1 \]
This gives us:
- \(xy \cdot x^2 = x^3y\)
- \(xy \cdot 1 = xy\)
- \(-2 \cdot x^2 = -2x^2\)
- \(-2 \cdot 1 = -2\)
Now we combine these results:
\[ x^3y + xy - 2x^2 - 2 \]
To rearrange it, we can write it in the standard polynomial form:
\[ x^3y - 2x^2 + xy - 2 \]
Thus, we can summarize our result where we place the like terms in a structured manner:
\[ x^3y + \underbrace{-2}{\text{constant term}} \underbrace{x^2}{\text{coefficient is } -2} + \underbrace{1}_{\text{coefficient of } xy}xy \]
However, we will identify the coefficients directly in order. Hence, we have:
The simplified expression is:
\[ x^3y - 2x^2 + xy - 2 \]
To finalize the answer, we just need to provide the values in the boxes as requested by the problem.
The completed expression is:
\[ x^{3}y + (-2)x^{2} + (1)xy - 2 \]
So the values to fill in the boxes are:
\(-2\) for the coefficient of \(x^2\) and \(1\) for the coefficient of \(xy\).
Thus, the final answer is:
\[ \boxed{-2} \text{ and } \boxed{1} \]