(-2sqrt (2)-2sqrt (2))/(2sqrt (3)-(2sqrt (2))/(2))

To simplify the given expression, let's first simplify the numerator and the denominator separately.<br /><br />The numerator is $-2\sqrt{2} - 2\sqrt{2}$. We can combine the like terms:<br /><br />$-2\sqrt{2} - 2\sqrt{2} = -4\sqrt{2}$<br /><br />So, the numerator simplifies to $-4\sqrt{2}$.<br /><br />Now, let's simplify the denominator. The denominator is $2\sqrt{3} - \frac{2\sqrt{2}}{2}$. We can simplify the second term in the denominator:<br /><br />$\frac{2\sqrt{2}}{2} = \sqrt{2}$<br /><br />So, the denominator simplifies to $2\sqrt{3} - \sqrt{2}$.<br /><br />Now, we can rewrite the expression as:<br /><br />$\frac{-4\sqrt{2}}{2\sqrt{3} - \sqrt{2}}$<br /><br />To simplify this expression further, we can multiply the numerator and the denominator by the conjugate of the denominator, which is $2\sqrt{3} + \sqrt{2}$:<br /><br />$\frac{-4\sqrt{2}}{2\sqrt{3} - \sqrt{2}} \times \frac{2\sqrt{3} + \sqrt{2}}{2\sqrt{3} + \sqrt{2}}$<br /><br />In the numerator, we have:<br /><br />$-4\sqrt{2} \times (2\sqrt{3} + \sqrt{2}) = -8\sqrt{6} - 4\sqrt{4} = -8\sqrt{6} - 8$<br /><br />In the denominator, we have:<br /><br />$(2\sqrt{3} - \sqrt{2})(2\sqrt{3} + \sqrt{2}) = (2\sqrt{3})^2 - (\sqrt{2})^2 = 4 \times 3 - 2 = 12 - 2 = 10$<br /><br />So, the expression simplifies to:<br /><br />$\frac{-8\sqrt{6} - 8}{10}$<br /><br />We can simplify this further by dividing both the numerator and the denominator by 2:<br /><br />$\frac{-4\sqrt{6} - 4}{5}$<br /><br />Therefore, the simplified form of the given expression is $\frac{-4\sqrt{6} - 4}{5}$.