1. **State the problem:** We need to find the expression for the function $$F(t) = (t^3 - 2t + 1)(2t^2 + 3t)$$ by expanding and simplifying it. 2. **Formula used:** To multiply two polynomials, use the distributive property (also called FOIL for binomials), which means multiply each term in the first polynomial by each term in the second polynomial. 3. **Multiply each term:** $$t^3 \times 2t^2 = 2t^5$$ $$t^3 \times 3t = 3t^4$$ $$-2t \times 2t^2 = -4t^3$$ $$-2t \times 3t = -6t^2$$ $$1 \times 2t^2 = 2t^2$$ $$1 \times 3t = 3t$$ 4. **Write the expanded form:** $$F(t) = 2t^5 + 3t^4 - 4t^3 - 6t^2 + 2t^2 + 3t$$ 5. **Combine like terms:** $$-6t^2 + 2t^2 = -4t^2$$ 6. **Final simplified expression:** $$F(t) = 2t^5 + 3t^4 - 4t^3 - 4t^2 + 3t$$ This is the expanded and simplified form of the function.