To solve the equation \[ \frac{(2\sqrt{3}-\sqrt{2})}{(\sqrt{3}+2\sqrt{2})} = m + n\sqrt{6}, \] we need to simplify the left-hand side and express it in the form \(m + n\sqrt{6}\). ### Step 1: Rationalizing the Denominator To simplify the fraction, we can multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \((\sqrt{3} + 2\sqrt{2})\) is \((\sqrt{3} - 2\sqrt{2})\). Thus, we have: \[ \frac{(2\sqrt{3}-\sqrt{2})(\sqrt{3}-2\sqrt{2})}{(\sqrt{3}+2\sqrt{2})(\sqrt{3}-2\sqrt{2})}. \] ### Step 2: Simplifying the Denominator The denominator simplifies as follows: \[ (\sqrt{3})^2 - (2\sqrt{2})^2 = 3 - 8 = -5. \] ### Step 3: Simplifying the Numerator Now, we simplify the numerator: \[ (2\sqrt{3}-\sqrt{2})(\sqrt{3}-2\sqrt{2}) = 2\sqrt{3} \cdot \sqrt{3} - 2\sqrt{3} \cdot 2\sqrt{2} - \sqrt{2} \cdot \sqrt{3} + \sqrt{2} \cdot 2\sqrt{2}. \] Calculating each term: 1. \(2\sqrt{3} \cdot \sqrt{3} = 2 \cdot 3 = 6\), 2. \(-2\sqrt{3} \cdot 2\sqrt{2} = -4\sqrt{6}\), 3. \(-\sqrt{2} \cdot \sqrt{3} = -\sqrt{6}\), 4. \(\sqrt{2} \cdot 2\sqrt{2} = 2 \cdot 2 = 4\). Combining these results, we have: \[ 6 - 4\sqrt{6} - \sqrt{6} + 4 = 10 - 5\sqrt{6}. \] ### Step 4: Putting It All Together Now, substituting back into our fraction, we get: \[ \frac{10 - 5\sqrt{6}}{-5} = -2 + \sqrt{6}. \] This can be rewritten as: \[ -2 + 1\sqrt{6}. \] ### Step 5: Identifying \(m\) and \(n\) From the expression \(-2 + 1\sqrt{6}\), we can identify: - \(m = -2\) - \(n = 1\) ### Conclusion Thus, the values of \(m\) and \(n\) are: \[ m = -2, \quad n = 1. \] The correct option is **B. -2, 1**. ### Explanation of Other Options - **Option A (1, -2)**: This is incorrect because the values of \(m\) and \(n\) do not match our derived values. - **Option C \(\left(\frac{-2}{5}, 1\right)\)**: This is incorrect because \(m\) is not \(\frac{-2}{5}\); it is \(-2\). - **Option D (2, \frac{3}{5})**: This is incorrect as neither \(m\) nor \(n\) matches the derived values. ### Revision Summary - Rationalize the denominator by multiplying by the conjugate. - Simplify both the numerator and denominator carefully. - Combine like terms to express the result in the form \(m + n\sqrt{6}\). - Identify \(m\) and \(n\) from the simplified expression.