1. The problem is to factorize $q^3 - 4^3$. 2. Recognize that this is a difference of cubes, which has the formula $$a^3 - b^3 = (a-b)(a^2 + ab + b^2).$$ 3. Using $a = q$ and $b = 4$, apply the formula: $$q^3 - 4^3 = (q - 4)(q^2 + 4q + 16).$$ 4. This is the factorized form of the expression. Final answer: $$q^3 - 4^3 = (q - 4)(q^2 + 4q + 16).$$