The expression provided in the image is:

$\left( -5.5 \div \frac{1}{4} \right) + \left( \frac{1}{2} - 3.8 \right) \cdot 4^2 - |3.2 - 6.5|$

Let’s break it down and solve it step by step:

1. First term: $-5.5 \div \frac{1}{4}$

   • Dividing by a fraction is the same as multiplying by its reciprocal: $-5.5 \div \frac{1}{4} = -5.5 \times 4 = -22$

2. Second term inside parentheses:

   • Calculate $\frac{1}{2} - 3.8$: $\frac{1}{2} = 0.5 \quad \text{so} \quad 0.5 - 3.8 = -3.3$

3. Third term (Multiplication and power):

   • We need to square $4$: $4^2 = 16$
   • Then multiply by the result from the second term: $(-3.3) \cdot 16 = -52.8$

4. Absolute value term:

   • Calculate $3.2 - 6.5$: $3.2 - 6.5 = -3.3$
   • The absolute value of $-3.3$ is $3.3$.

Now, let's put everything together:

$-22 + (-52.8) - 3.3$

Finally, let's calculate the result.The final result of the expression is $-78.1$.

Would you like further details or have any questions about the steps? Here are a few related questions to explore:

1. What are the properties of division and multiplication when dealing with fractions?
2. How does the order of operations affect solving complex expressions like this one?
3. Why do we take the absolute value of a number, and how does it impact the calculation?
4. How does the distributive property play a role in expressions involving parentheses?
5. How can complex expressions like this be simplified more efficiently?

Tip: Always ensure you follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) to avoid errors in solving such expressions.