1. **State the problem:** Expand and fully simplify the expression $$(x^2 + 3x + 2)(x + 8)$$. 2. **Formula and rules:** To expand, 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. **Expand step-by-step:** - Multiply $x^2$ by $x$ and $8$: $$x^2 \cdot x = x^3$$ and $$x^2 \cdot 8 = 8x^2$$ - Multiply $3x$ by $x$ and $8$: $$3x \cdot x = 3x^2$$ and $$3x \cdot 8 = 24x$$ - Multiply $2$ by $x$ and $8$: $$2 \cdot x = 2x$$ and $$2 \cdot 8 = 16$$ 4. **Combine all terms:** $$x^3 + 8x^2 + 3x^2 + 24x + 2x + 16$$ 5. **Simplify by combining like terms:** $$x^3 + (8x^2 + 3x^2) + (24x + 2x) + 16 = x^3 + 11x^2 + 26x + 16$$ 6. **Final answer:** $$x^3 + 11x^2 + 26x + 16$$