27. Find the value of
$$
\frac{x^{4}-6 x^{3}-2 x^{2}+18 x+23}{x^{2}-8 x+15}
$$
when $x=\sqrt{19-8 \sqrt{3}}$.
27. Ans: 5 .
Observe $x=\sqrt{(4-\sqrt{3})^{2}}=4-\sqrt{3}$ and $x^{2}-8 x+13=19-8 \sqrt{3}-32+8 \sqrt{3}+13=0$. Use long division,
$$
x^{4}-6 x^{3}-2 x^{2}+18 x+23=\left(x^{2}-8 x+13\right)\left(x^{2}+2 x+1\right)+10 .
$$

Thus the original expression equals
$$
\frac{\left(x^{2}-8 x+13\right)\left(x^{2}+2 x+1\right)+10}{x^{2}-8 x+13+2}=\frac{10}{2}=5 .
$$