1. **State the problem:** Simplify the expression $$(2x^2 - 2x + 5)(x + 4)$$ and write it in the form $$ax^3 + bx^2 + cx + d$$. 2. **Use the distributive property (FOIL for polynomials):** Multiply each term in the first polynomial by each term in the second polynomial. 3. **Multiply terms:** $$2x^2 \cdot x = 2x^3$$ $$2x^2 \cdot 4 = 8x^2$$ $$-2x \cdot x = -2x^2$$ $$-2x \cdot 4 = -8x$$ $$5 \cdot x = 5x$$ $$5 \cdot 4 = 20$$ 4. **Combine like terms:** $$2x^3 + (8x^2 - 2x^2) + (-8x + 5x) + 20$$ 5. **Simplify:** $$2x^3 + 6x^2 - 3x + 20$$ **Final answer:** $$2x^3 + 6x^2 - 3x + 20$$