The problem involves evaluating a complex mathematical expression. I will transcribe and solve the problem step by step.

Transcription of the problem:

The expression appears as follows:

$4 - \frac{5}{\left( 3 - \frac{5}{\left( \frac{28}{14} - 7 \right)} \right)} + \frac{9}{\left( 14 - \frac{3}{\left( 4 - 2 \right)} \right)}$

Step-by-step solution:

1. Simplify the innermost fractions:

• $\frac{28}{14} = 2$,
• $\frac{3}{4 - 2} = \frac{3}{2}$.

2. Substitute back into the expression:

$4 - \frac{5}{\left( 3 - \frac{5}{2 - 7} \right)} + \frac{9}{\left( 14 - \frac{3}{\frac{3}{2}} \right)}.$

3. Simplify further:

• $2 - 7 = -5$,
• $\frac{5}{-5} = -1$,
• $3 - (-1) = 4$,
• For the second fraction, $\frac{3}{\frac{3}{2}} = 2$, and $14 - 2 = 12$.

Substitute these values: $4 - \frac{5}{4} + \frac{9}{12}.$

4. Simplify the fractions:

• $\frac{5}{4}$ remains as is,
• $\frac{9}{12} = \frac{3}{4}$.

The expression becomes: $4 - \frac{5}{4} + \frac{3}{4}.$

5. Combine terms:

Write $4$ as $\frac{16}{4}$ to match the denominator: $\frac{16}{4} - \frac{5}{4} + \frac{3}{4}.$

Combine: $\frac{16 - 5 + 3}{4} = \frac{14}{4} = \frac{7}{2}.$

Final Answer:

$\boxed{\frac{7}{2}}$

Would you like further clarification or additional steps broken down? Here are related questions:

1. How do you simplify fractions within nested expressions?
2. What are some common mistakes to avoid when solving problems like this?
3. How can fraction simplifications speed up calculations?
4. What methods are useful for verifying your work in complex expressions?
5. How does proper grouping affect the solution of nested arithmetic?

Tip: Always work step by step and handle the innermost operations first when dealing with nested expressions.