$x = 5-2\sqrt3 \implies (x-5)^2 = 12 \implies x^2 - 10x = -13$

As the hint says, we have $x^2 = 10x-13$. Then the denominator becomes $-13 + 19 = 6$. The numerator, we can use the following equations, because $x \neq 0$: \begin{align*} x^2 &= 10x - 13 \\ x^3 &= 10x^2 - 13x \\ x^4 &= 10x^3 - 13x^2\end{align*} Then, we can simplify the numerator to $x^3 - 8 x^2 - 7x + 68$ and then $2x^2 - 20x + 68$ which is $2(x^2-10x+34) = 2(-13+34) = 42$. Therefore the answer is $\dfrac{42}{6} = \boxed{7}$