Let $x=4+\frac{1}{5+\frac{1}{4+\frac{1}{5+\cdots}}}$ Since the pattern repeats after the first term, the expression after the first $4+\cdots$ is again related to $x$.

1. Set up the repeating part

Write $x=4+\frac{1}{5+\frac{1}{x}}$ because after $5+\frac{1}{\cdots}$, the infinite tail is again the same whole expression $x$.

2. Simplify the equation

So, $x-4=\frac{1}{5+\frac{1}{x}}$ Now simplify the denominator: $5+\frac{1}{x}=\frac{5x+1}{x}$ Hence, $x-4=\frac{1}{\frac{5x+1}{x}}=\frac{x}{5x+1}$

3. Form the quadratic equation

Thus, $(x-4)(5x+1)=x$ Expand: $5x^2+x-20x-4=x$ $5x^2-19x-4=x$ $5x^2-20x-4=0$ Divide by $1$ and simplify by factoring out nothing further: $5x^2-20x-4=0$

4. Solve the quadratic

Using the quadratic formula, $x=\frac{20\pm\sqrt{(-20)^2-4(5)(-4)}}{2\cdot 5}$ $x=\frac{20\pm\sqrt{400+80}}{10}$ $x=\frac{20\pm\sqrt{480}}{10}$ $x=\frac{20\pm 4\sqrt{30}}{10}$ $x=2\pm \frac{2}{5}\sqrt{30}$

5. Choose the valid value

Since the given continued fraction is clearly greater than $4$, we must take the positive root: $x=2+\frac{2}{5}\sqrt{30}$

6. Compare with options

Option A is: $2+\frac{2}{5}\sqrt{30}$ So the correct option is A.

7. Compare with stored correct answer

Stored correct answer: A

Our derived answer matches the stored answer.