Answer:
To simplify the given expression, let's rationalize the denominator first. We have the expression:
\(\frac{2\sqrt{5} + 3\sqrt{2}}{2\sqrt{3} - 3\sqrt{2}}\)
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator, which is \(2\sqrt{3} + 3\sqrt{2}\):
\(\frac{(2\sqrt{5} + 3\sqrt{2})(2\sqrt{3} + 3\sqrt{2})}{(2\sqrt{3} - 3\sqrt{2})(2\sqrt{3} + 3\sqrt{2})}\)
Expanding the numerator and denominator, we get:
\(\frac{4\sqrt{15} + 6\sqrt{10} + 6\sqrt{10} + 9\sqrt{4}}{4\sqrt{9} - 9\sqrt{4}}\)
Simplifying further, we have:
\(\frac{4\sqrt{15} + 12\sqrt{10} + 6 \cdot 2}{4 \cdot 3 - 9 \cdot 2}\)
\(\frac{4\sqrt{15} + 12\sqrt{10} + 12}{12 - 18}\)
\(\frac{4\sqrt{15} + 12\sqrt{10} + 12}{-6}\)
Now, we can simplify the expression and rewrite it as:
\(-\frac{2\sqrt{15} + 6\sqrt{10} + 6}{3}\)
Comparing this with \(m + n\sqrt{6}\), we can determine that \(m = -2\) and \(n = -6\). Therefore, the value of \(m\) is -2 and the value of \(n\) is -6.