Multiply. \( (4 \sqrt{3}-9 \sqrt{2})(\sqrt{3}+6 \sqrt{2}) \) \( (4 \sqrt{3}-9 \sqrt{2})(\sqrt{3}+6 \sqrt{2})=\square \) (Type an exact answer, using radicals as needed. Simplify your answer.)

To multiply the expression \( (4 \sqrt{3}-9 \sqrt{2})(\sqrt{3}+6 \sqrt{2}) \), we will use the distributive property (also known as the FOIL method for binomials). 1. **First**: \(4 \sqrt{3} \times \sqrt{3} = 4 \times 3 = 12\). 2. **Outer**: \(4 \sqrt{3} \times 6 \sqrt{2} = 24 \sqrt{6}\). 3. **Inner**: \(-9 \sqrt{2} \times \sqrt{3} = -9 \sqrt{6}\). 4. **Last**: \(-9 \sqrt{2} \times 6 \sqrt{2} = -54\). Now, combine all these results: \[ (4 \sqrt{3}-9 \sqrt{2})(\sqrt{3}+6 \sqrt{2}) = 12 + 24\sqrt{6} - 9\sqrt{6} - 54 \] Combine like terms: \[ 12 - 54 + (24\sqrt{6} - 9\sqrt{6}) = -42 + 15\sqrt{6} \] Thus, the final simplified answer is: \[ (4 \sqrt{3}-9 \sqrt{2})(\sqrt{3}+6 \sqrt{2}) = -42 + 15\sqrt{6}. \]