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