1. **State the problem:** Find the product of the polynomials $$(x^2 - x^3 + 1)(x^2 - 1).$$ 2. **Recall the distributive property:** To multiply two polynomials, multiply each term in the first polynomial by each term in the second polynomial. 3. **Multiply each term:** $$x^2 \cdot x^2 = x^4$$ $$x^2 \cdot (-1) = -x^2$$ $$-x^3 \cdot x^2 = -x^5$$ $$-x^3 \cdot (-1) = +x^3$$ $$1 \cdot x^2 = x^2$$ $$1 \cdot (-1) = -1$$ 4. **Write the expanded expression:** $$x^4 - x^2 - x^5 + x^3 + x^2 - 1$$ 5. **Combine like terms:** Note that $-x^2$ and $+x^2$ cancel out: $$x^4 - \cancel{x^2} - x^5 + x^3 + \cancel{x^2} - 1 = -x^5 + x^4 + x^3 - 1$$ 6. **Final answer:** $$\boxed{-x^5 + x^4 + x^3 - 1}$$