1. **State the problem:** Fully factorize the expression $x^2 + 12x$. 2. **Identify the common factor:** Both terms $x^2$ and $12x$ have a common factor of $x$. 3. **Factor out the common factor:** $$x^2 + 12x = x(x + 12)$$ 4. **Check if further factorization is possible:** The binomial $x + 12$ cannot be factored further since it is a sum of unlike terms. 5. **Final answer:** The fully factorized form is $$x(x + 12)$$ This means we have expressed the original quadratic as a product of two factors, which is the goal of factorization.