To find the quotient \(\frac{\sqrt{6} + \sqrt{11}}{\sqrt{5} + \sqrt{3}}\), we need to simplify the given expression.
The quotient given is:
[tex]\[
\frac{\sqrt{6} + \sqrt{11}}{\sqrt{5} + \sqrt{3}}
\][/tex]
Looking at the possible answers provided:
1. \(\frac{\sqrt{30} + 3\sqrt{2} + \sqrt{55} + \sqrt{33}}{8}\)
2. \(\frac{\sqrt{30} - 3\sqrt{2} + \sqrt{55} - \sqrt{33}}{2}\)
3. \(\frac{17}{8}\)
4. \(-\frac{5}{2}\)
We compare our original expression to see if it matches any of the given choices.
Our original quotient is simply:
[tex]\[
\frac{\sqrt{6} + \sqrt{11}}{\sqrt{5} + \sqrt{3}}
\][/tex]
Notice that it directly represents the simplified form of the fraction where the numerator is \(\sqrt{6} + \sqrt{11}\) and the denominator is \(\sqrt{5} + \sqrt{3}\).
Thus, the correct answer is:
[tex]\[
\frac{\sqrt{6} + \sqrt{11}}{\sqrt{5} + \sqrt{3}}
\][/tex]
Therefore, none of the additional complex expressions (choices 1 and 2) or the plain fractions (choices 3 and 4) match the form of the quotient we're looking for. Hence, the result is exactly as presented in the original quotient:
[tex]\[
\boxed{\frac{\sqrt{6} + \sqrt{11}}{\sqrt{5} + \sqrt{3}}}
\][/tex]
