ANSWER
[tex] \sqrt{11} - 4[/tex]
EXPLANATION
The given radical expression is
[tex] \frac{3 - 2 \sqrt{11} }{2 + \sqrt{11} } [/tex]
We rationalize the radical expression by multiplying both the numerator and denominator by the conjugate of
[tex]2 + \sqrt{11} [/tex]
which is
[tex]2 - \sqrt{11} [/tex]
This will give us,
[tex] \frac{3 - 2 \sqrt{11} }{2 + \sqrt{11} } \times \frac{2 - \sqrt{11} }{2 - \sqrt{11} } [/tex]
[tex] \frac{(3 - 2 \sqrt{11})(2 - \sqrt{11} ) }{(2 + \sqrt{11})(2 - \sqrt{11} ) }[/tex]
We expand the numerator and also apply difference of two squares to the denominator.
[tex] \frac{6 - 3 \sqrt{11} - 4 \sqrt{11} + 2(11)}{ {2}^{2} - { (\sqrt{11} )}^{2} } [/tex]
This simplifies to,
[tex] \frac{6- 3\sqrt{11} - 4 \sqrt{11} + 22}{ 4- 11} [/tex]
[tex] \frac{28- 7 \sqrt{11} }{ - 7} [/tex]
We now share the denominator for the numerators to obtain,
[tex] - \frac{28}{7} + \frac{7 \sqrt{11} }{7} [/tex]
This finally gives,
[tex] - 4 + \sqrt{11} = \sqrt{11} - 4[/tex]
