1. **State the problem:** Simplify the expression $$6 \cdot (-a^6 b)^2 + 9a^{12} b^2$$. 2. **Recall the formula and rules:** When raising a power to another power, multiply the exponents: $$(x^m)^n = x^{mn}$$. 3. **Simplify the term $(-a^6 b)^2$:** $$(-a^6 b)^2 = (-1)^2 \cdot (a^6)^2 \cdot b^2 = 1 \cdot a^{12} \cdot b^2 = a^{12} b^2$$ 4. **Multiply by 6:** $$6 \cdot a^{12} b^2 = 6a^{12} b^2$$ 5. **Add the terms:** $$6a^{12} b^2 + 9a^{12} b^2 = (6 + 9) a^{12} b^2 = 15a^{12} b^2$$ 6. **Final answer:** $$\boxed{15a^{12} b^2}$$