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Let's simplify the given expression step by step.

The expression provided is:

\[
\frac{2^{1117}(2 - 2^{2874}) - (3 + 3i)}{(2 + 5i) + (2 + 5i)}
\]

### Step 1: Simplify the denominator
First, simplify the denominator:

\[
(2 + 5i) + (2 + 5i) = 2 + 5i + 2 + 5i = 4 + 10i
\]

So the expression becomes:

\[
\frac{2^{1117}(2 - 2^{2874}) - (3 + 3i)}{4 + 10i}
\]

### Step 2: Simplify the numerator
Next, simplify the numerator:

\[
2^{1117}(2 - 2^{2874}) - (3 + 3i)
\]

Expand the term \(2^{1117}(2 - 2^{2874})\):

\[
2^{1117} \times 2 - 2^{1117} \times 2^{2874} = 2^{1118} - 2^{3991}
\]

So the numerator is:

\[
2^{1118} - 2^{3991} - (3 + 3i)
\]

### Step 3: Combine terms
Now the expression is:

\[
\frac{2^{1118} - 2^{3991} - 3 - 3i}{4 + 10i}
\]

### Step 4: Simplify the complex fraction
To simplify this expression further, multiply the numerator and denominator by the complex conjugate of the denominator to remove the imaginary part from the denominator.

The complex conjugate of \(4 + 10i\) is \(4 - 10i\).

Multiply both the numerator and denominator by \(4 - 10i\):

\[
\frac{(2^{1118} - 2^{3991} - 3 - 3i)(4 - 10i)}{(4 + 10i)(4 - 10i)}
\]

### Step 5: Calculate the denominator
First, calculate the denominator:

\[
(4 + 10i)(4 - 10i) = 4^2 - (10i)^2 = 16 - 100(-1) = 16 + 100 = 116
\]

So the denominator becomes \(116\).

### Step 6: Expand the numerator
Now expand the numerator:

\[
(2^{1118} - 2^{3991} - 3 - 3i)(4 - 10i)
\]

Expanding this expression:

\[
= 2^{1118} \times 4 - 2^{1118} \times 10i - 2^{3991} \times 4 + 2^{3991} \times 10i - 3 \times 4 + 3 \times 10i - 3i \times 4 + 3i \times 10i
\]

Simplify further:

\[
= 4 \times (2^{1118} - 2^{3991}) - 10i \times (2^{1118} - 2^{3991}) - 12 + 30i - 12i - 30
\]

Combine like terms to simplify:

\[
= 4(2^{1118} - 2^{3991}) - 42 - (10i(2^{1118} - 2^{3991}) + 18i)
\]

### Step 7: Final Expression
So, the final expression is:

\[
\frac{4(2^{1118} - 2^{3991}) - 42 - (10i(2^{1118} - 2^{3991}) + 18i)}{116}
\]

This is the simplified form, but depending on further calculations or factorizations, you might reduce the expression more. However, in terms of algebraic simplification, this is the core process.

Would you like further details on any step, or do you have any questions?

### Follow-up Questions:
1. How does multiplying by the conjugate help in simplifying complex fractions?
2. Can the expression \(2^{1118} - 2^{3991}\) be factored further?
3. What are the benefits of writing the denominator in its simplest form?
4. How would the final expression change if \(i\) was treated differently?
5. What is the significance of using \(i\) in complex number simplifications?

### Tip:
Always check if complex numbers can be simplified by converting them into a modulus-argument form for easier multiplication and division.

Learn how to simplify the complex fraction involving 2^{1117}(2 - 2^{2874}) and complex numbers step-by-step. This problem involves advanced concepts in complex number arithmetic and fraction simplification.

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