1. **Problem:** Simplify the algebraic expression $3a(2a - 3b + 4c)$ correctly. 2. **Formula and rules:** Use the distributive property: $a(b + c) = ab + ac$. 3. **Step-by-step solution:** 1. Apply distributive property: $$3a(2a - 3b + 4c) = 3a \times 2a - 3a \times 3b + 3a \times 4c$$ 2. Multiply each term: $$= 6a^2 - 9ab + 12ac$$ 4. **Explanation:** Multiply the coefficient and variables carefully. $3a \times 2a$ gives $6a^2$ because $a \times a = a^2$. Similarly, multiply coefficients and variables for other terms. 5. **Final answer:** $$6a^2 - 9ab + 12ac$$