Multiply. \[ (4 \sqrt{10}+7 \sqrt{3})(3 \sqrt{3}-4 \sqrt{10}) \] Simplify your answer as much as possible. \( \square \)

To multiply the two expressions \( (4 \sqrt{10} + 7 \sqrt{3})(3 \sqrt{3} - 4 \sqrt{10}) \), we can use the distributive property (also known as the FOIL method for binomials). First, distribute each term in the first parentheses to each term in the second parentheses: 1. \( 4 \sqrt{10} \cdot 3 \sqrt{3} = 12 \sqrt{30} \) 2. \( 4 \sqrt{10} \cdot (-4 \sqrt{10}) = -16 \cdot 10 = -160 \) 3. \( 7 \sqrt{3} \cdot 3 \sqrt{3} = 21 \cdot 3 = 63 \) 4. \( 7 \sqrt{3} \cdot (-4 \sqrt{10}) = -28 \sqrt{30} \) Now, combine these results: \[ 12 \sqrt{30} - 160 + 63 - 28 \sqrt{30} \] Next, combine like terms. The terms with \( \sqrt{30} \) are: \[ 12 \sqrt{30} - 28 \sqrt{30} = -16 \sqrt{30} \] And the constant terms: \[ -160 + 63 = -97 \] Putting it all together, we have: \[ -16 \sqrt{30} - 97 \] Thus, the simplified answer is: \[ \boxed{-16 \sqrt{30} - 97} \]