1. **State the problem:** Simplify the expression $$(b^2 + 1)(-b) + (-b + 1)(1 - b^2).$$ 2. **Recall distributive property:** To simplify, we will expand each product using the distributive property: $$a(b+c) = ab + ac.$$ Also, remember that multiplying by $-1$ changes the sign. 3. **Expand the first product:** $$(b^2 + 1)(-b) = b^2 \cdot (-b) + 1 \cdot (-b) = -b^3 - b.$$ 4. **Expand the second product:** $$(-b + 1)(1 - b^2) = (-b)(1) + (-b)(-b^2) + 1 \cdot 1 + 1 \cdot (-b^2) = -b + b^3 + 1 - b^2.$$ 5. **Combine all terms:** $$-b^3 - b + (-b + b^3 + 1 - b^2) = -b^3 - b - b + b^3 + 1 - b^2.$$ 6. **Group like terms:** $$(-b^3 + b^3) + (-b - b) + (-b^2) + 1 = 0 - 2b - b^2 + 1.$$ 7. **Final simplified expression:** $$1 - b^2 - 2b.$$ This is the simplified form of the original expression.