Multiply. Assume that all variables represent positive real numbers. \[ (7 \sqrt{x}+6)(8 \sqrt{x}+4) \] \( (7 \sqrt{x}+6)(8 \sqrt{x}+4)=\square \) (Simplify your answer. Type an exact answer, using radicals as needed.

To multiply the expression \( (7 \sqrt{x}+6)(8 \sqrt{x}+4) \), we can use the distributive property (also known as the FOIL method for binomials): \[ (7 \sqrt{x} + 6)(8 \sqrt{x} + 4) = 7 \sqrt{x} \cdot 8 \sqrt{x} + 7 \sqrt{x} \cdot 4 + 6 \cdot 8 \sqrt{x} + 6 \cdot 4. \] Calculating each part: 1. \( 7 \sqrt{x} \cdot 8 \sqrt{x} = 56x \) (since \(\sqrt{x} \cdot \sqrt{x} = x\)), 2. \( 7 \sqrt{x} \cdot 4 = 28 \sqrt{x} \), 3. \( 6 \cdot 8 \sqrt{x} = 48 \sqrt{x} \), 4. \( 6 \cdot 4 = 24 \). Now, combine all the terms: \[ 56x + 28\sqrt{x} + 48\sqrt{x} + 24. \] Combine the like terms: \[ 56x + (28\sqrt{x} + 48\sqrt{x}) + 24 = 56x + 76\sqrt{x} + 24. \] Thus, the final simplified expression is: \[ (7 \sqrt{x}+6)(8 \sqrt{x}+4) = 56x + 76\sqrt{x} + 24. \]