1. **State the problem:** Simplify the product $\left(\sqrt{5} + 6\right)\left(\sqrt{5} - 6\right)$.\n\n2. **Formula used:** This is a product of conjugates, which follows the difference of squares formula: $$(a+b)(a-b) = a^2 - b^2.$$\n\n3. **Identify terms:** Here, $a = \sqrt{5}$ and $b = 6$.\n\n4. **Apply the formula:**\n$$\left(\sqrt{5} + 6\right)\left(\sqrt{5} - 6\right) = (\sqrt{5})^2 - 6^2.$$\n\n5. **Calculate squares:**\n$$(\sqrt{5})^2 = 5,$$\n$$6^2 = 36.$$\n\n6. **Substitute back:**\n$$5 - 36 = -31.$$\n\n7. **Final answer:** The product simplifies to $\boxed{-31}$.