Multiply and Simplify: \( (5 \sqrt{2}+\sqrt{3})(\sqrt{2}-2 \sqrt{3}) \)

To multiply and simplify \( (5 \sqrt{2}+\sqrt{3})(\sqrt{2}-2 \sqrt{3}) \), we can use the distributive property (also known as the FOIL method for binomials). 1. **First Terms**: \( 5 \sqrt{2} \cdot \sqrt{2} = 5 \cdot 2 = 10 \). 2. **Outer Terms**: \( 5 \sqrt{2} \cdot (-2 \sqrt{3}) = -10 \sqrt{6} \). 3. **Inner Terms**: \( \sqrt{3} \cdot \sqrt{2} = \sqrt{6} \). 4. **Last Terms**: \( \sqrt{3} \cdot (-2 \sqrt{3}) = -2 \cdot 3 = -6 \). Now we combine all these results: \[ 10 - 10 \sqrt{6} + \sqrt{6} - 6 \] Combining like terms: - The constant terms: \( 10 - 6 = 4 \) - The terms with \( \sqrt{6} \): \( -10 \sqrt{6} + \sqrt{6} = -9 \sqrt{6} \) Putting it all together, we have: \[ 4 - 9 \sqrt{6} \] Thus, the simplified result is: \[ \boxed{4 - 9 \sqrt{6}} \]