Let's solve the expression step by step.

Given expression: $\frac{2}{10} \times (6 + 8 \times 3 - 2) + \left[\frac{1}{5} + \frac{7}{25} - \left(\frac{3}{7} + \frac{7}{14}\right)\right]$

1. Simplify inside the parentheses:

   • First, calculate $8 \times 3 = 24$.
   • Then, $6 + 24 - 2 = 28$.

2. Multiply by $\frac{2}{10}$: $\frac{2}{10} \times 28 = \frac{56}{10} = 5.6$

3. Simplify inside the brackets:

   • Simplify the fraction inside the curly braces: $\frac{7}{14} = \frac{1}{2} = 0.5$ So, \text{(Finding a common denominator here)} = \frac{6}{14} + \frac{7}{14} = \frac{13}{14}$$
   • Now the bracket part becomes: $\frac{1}{5} + \frac{7}{25} - \frac{13}{14}$
     • Convert each to a common denominator or simplify directly: $\text{which is } = \frac{5}{25} + \frac{7}{25} = \frac{12}{25}$

4. Subtract:

   • Subtract the fractions in the square bracket:
   • $\frac{12}{25} - \frac{13}{14} = - 1.5$

5. Final Addition:

   • Add the results: $5.6 - 1.5 = 4.1$

So the direct answer is 4.1.

Would you like a more detailed breakdown or have any questions? Here are some related questions you might consider:

1. What is the method to simplify expressions with multiple operations?
2. How do you find a common denominator between fractions?
3. What is the significance of parentheses in mathematical expressions?
4. How can you simplify complex fractions?
5. What is the order of operations in mathematics?
6. How do you handle negative results when subtracting fractions?
7. How do you multiply a fraction by an integer?
8. What are some tips for avoiding mistakes in multi-step calculations?

Tip: Always simplify the fractions and terms inside parentheses first before proceeding with multiplication or addition outside.