The given image contains a mathematical expression that needs to be simplified, with four possible answer choices:

$4 - \frac{5}{1 + \frac{1}{3 + \frac{1}{2 + \frac{1}{4}}}}$

Let's break this expression down step by step, starting with the innermost fraction.

1. Simplify $2 + \frac{1}{4}$: $2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4}$

2. Substitute into the next part: $3 + \frac{1}{\frac{9}{4}} = 3 + \frac{4}{9} = \frac{27}{9} + \frac{4}{9} = \frac{31}{9}$

3. Now substitute this into the next part: $1 + \frac{1}{\frac{31}{9}} = 1 + \frac{9}{31} = \frac{31}{31} + \frac{9}{31} = \frac{40}{31}$

4. Substitute this into the outer fraction: $\frac{5}{\frac{40}{31}} = 5 \times \frac{31}{40} = \frac{155}{40} = \frac{31}{8}$

5. Finally, subtract this from 4: $4 - \frac{31}{8} = \frac{32}{8} - \frac{31}{8} = \frac{1}{8}$

Thus, the correct answer is $\frac{1}{8}$, which corresponds to option (a).

Would you like a deeper explanation or more details on any step? Here are some additional questions for practice:

1. What happens if we change the value of 4 in the expression to 5?
2. How would the result change if the fraction $\frac{1}{4}$ was replaced by $\frac{1}{5}$?
3. Can you generalize a method for simplifying such nested fractions?
4. What would the expression become if all fractions were halved?
5. How does the method of continued fractions help in simplifying such problems?

Tip: Always start simplifying the innermost part of the fraction and work your way outwards for complex nested expressions.