1. **State the problem:** Expand the expression $ (x+4)^3 $. 2. **Formula used:** The binomial expansion for $ (a+b)^3 $ is given by $$ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $$ 3. **Apply the formula:** Here, $a = x$ and $b = 4$. Substitute these values: $$ (x+4)^3 = x^3 + 3x^2(4) + 3x(4)^2 + 4^3 $$ 4. **Calculate each term:** $$ 3x^2(4) = 12x^2 $$ $$ 3x(4)^2 = 3x(16) = 48x $$ $$ 4^3 = 64 $$ 5. **Write the expanded form:** $$ (x+4)^3 = x^3 + 12x^2 + 48x + 64 $$ 6. **Final answer:** $$ \boxed{x^3 + 12x^2 + 48x + 64} $$