Given that \( x=\sqrt{3}-2 \), simplify \( \frac{x^{2}+1}{x^{2}-5} \) without using a calculator. (Give your answer in simplest surd form.)

To simplify \( \frac{x^{2}+1}{x^{2}-5} \) where \( x=\sqrt{3}-2 \), we begin by calculating \( x^2 \). First, calculate \( x^2 \): \[ x = \sqrt{3} - 2 \] \[ x^2 = (\sqrt{3} - 2)^2 = (\sqrt{3})^2 - 2 \times 2 \times \sqrt{3} + 2^2 = 3 - 4\sqrt{3} + 4 = 7 - 4\sqrt{3} \] Now, substitute \( x^2 \) into the expression \( \frac{x^2 + 1}{x^2 - 5} \): Calculating \( x^2 + 1 \): \[ x^2 + 1 = (7 - 4\sqrt{3}) + 1 = 8 - 4\sqrt{3} \] Now calculating \( x^2 - 5 \): \[ x^2 - 5 = (7 - 4\sqrt{3}) - 5 = 2 - 4\sqrt{3} \] The expression we want to simplify is now: \[ \frac{x^2 + 1}{x^2 - 5} = \frac{8 - 4\sqrt{3}}{2 - 4\sqrt{3}} \] Next, we can factor out a common term from the numerator and denominator: \[ = \frac{4(2 - \sqrt{3})}{2 - 4\sqrt{3}} = \frac{4(2 - \sqrt{3})}{2(1 - 2\sqrt{3})} \] Now, simplify: \[ = \frac{2(2 - \sqrt{3})}{1 - 2\sqrt{3}} \] For ease of simplification, we can multiply the numerator and denominator by the conjugate of the denominator: \[ \frac{2(2 - \sqrt{3})(1 + 2\sqrt{3})}{(1 - 2\sqrt{3})(1 + 2\sqrt{3})} \] Calculating the denominator: \[ (1 - 2\sqrt{3})(1 + 2\sqrt{3}) = 1^2 - (2\sqrt{3})^2 = 1 - 12 = -11 \] Now calculate the numerator: \[ 2(2 - \sqrt{3})(1 + 2\sqrt{3}) = 2[(2)(1) + (2)(2\sqrt{3}) - (\sqrt{3})(1) - (\sqrt{3})(2\sqrt{3})] \] \[ = 2[2 + 4\sqrt{3} - \sqrt{3} - 6] = 2[-4 + 3\sqrt{3}] \] \[ = -8 + 6\sqrt{3} \] Thus, we have: \[ \frac{-8 + 6\sqrt{3}}{-11} = \frac{8 - 6\sqrt{3}}{11} \] Putting it all together, the simplified form of \( \frac{x^2+1}{x^2-5} \) is: \[ \frac{8 - 6\sqrt{3}}{11} \] Therefore, the final answer in simplest surd form is: \[ \frac{8 - 6\sqrt{3}}{11} \]