1. **State the problem:** Simplify the expression $3xy \cdot 4xy^2$. 2. **Formula and rules:** When multiplying monomials, multiply the coefficients (numbers) and then multiply the variables by adding their exponents if they have the same base. 3. **Multiply coefficients:** $3 \times 4 = 12$. 4. **Multiply variables:** For $x$, we have $x^1 \times x^1 = x^{1+1} = x^2$. For $y$, we have $y^1 \times y^2 = y^{1+2} = y^3$. 5. **Combine results:** The product is $12x^2y^3$. 6. **Final answer:** $$12x^2y^3$$