Multiply. \( (8 \sqrt{5}+9 \sqrt{2})(9 \sqrt{5}+6 \sqrt{2}) \) Simplify your answer as much as possible.

To multiply the expressions \( (8 \sqrt{5}+9 \sqrt{2})(9 \sqrt{5}+6 \sqrt{2}) \), we can use the distributive property (also known as the FOIL method for binomials): \[ = 8 \sqrt{5} \cdot 9 \sqrt{5} + 8 \sqrt{5} \cdot 6 \sqrt{2} + 9 \sqrt{2} \cdot 9 \sqrt{5} + 9 \sqrt{2} \cdot 6 \sqrt{2} \] Calculating each term step by step, we have: 1. \( 8 \sqrt{5} \cdot 9 \sqrt{5} = 72 \cdot 5 = 360 \) 2. \( 8 \sqrt{5} \cdot 6 \sqrt{2} = 48 \sqrt{10} \) 3. \( 9 \sqrt{2} \cdot 9 \sqrt{5} = 81 \sqrt{10} \) 4. \( 9 \sqrt{2} \cdot 6 \sqrt{2} = 54 \cdot 2 = 108 \) Now, combining all these terms together: \[ = 360 + 48 \sqrt{10} + 81 \sqrt{10} + 108 \] Now, combine like terms: \[ = (360 + 108) + (48 \sqrt{10} + 81 \sqrt{10}) = 468 + 129 \sqrt{10} \] Thus, the simplified answer is: \[ \boxed{468 + 129 \sqrt{10}} \]