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obten la forma binomica del siguiente ejercicio

To solve the given expression and obtain its binomial form, let's break down each part.

The expression is:

\[
\frac{1+i}{1-i} = (1+2i)(2+2i) + \frac{3-i}{1+i}
\]

### Step 1: Simplify \(\frac{1+i}{1-i}\)
To simplify \(\frac{1+i}{1-i}\), multiply the numerator and the denominator by the conjugate of the denominator, \(1+i\):

\[
\frac{1+i}{1-i} \times \frac{1+i}{1+i} = \frac{(1+i)(1+i)}{(1-i)(1+i)} = \frac{1 + 2i + i^2}{1 - i^2} = \frac{1 + 2i - 1}{1 + 1} = \frac{2i}{2} = i
\]

So, \(\frac{1+i}{1-i} = i\).

### Step 2: Simplify \((1+2i)(2+2i)\)
Use the distributive property to expand:

\[
(1+2i)(2+2i) = 1 \cdot 2 + 1 \cdot 2i + 2i \cdot 2 + 2i \cdot 2i
\]

\[
= 2 + 2i + 4i + 4i^2
\]

Since \(i^2 = -1\):

\[
2 + 6i - 4 = -2 + 6i
\]

### Step 3: Simplify \(\frac{3-i}{1+i}\)
To simplify \(\frac{3-i}{1+i}\), multiply the numerator and the denominator by the conjugate of the denominator, \(1-i\):

\[
\frac{3-i}{1+i} \times \frac{1-i}{1-i} = \frac{(3-i)(1-i)}{(1+i)(1-i)} = \frac{3 - 3i - i + i^2}{1 - i^2} = \frac{3 - 4i - 1}{1 + 1} = \frac{2 - 4i}{2} = 1 - 2i
\]

### Step 4: Combine the results
Now, substitute the simplified expressions back into the original equation:

\[
i = (-2 + 6i) + (1 - 2i)
\]

\[
i = (-2 + 1) + (6i - 2i)
\]

\[
i = -1 + 4i
\]

### Final Answer:
The binomial form of the given expression is:

\[
\boxed{-1 + 5i}
\]

Would you like more details, or do you have any questions about this solution?

### Related Questions:
1. How do you multiply complex numbers in binomial form?
2. What is the significance of using the conjugate in simplifying complex fractions?
3. Can you explain the process of converting from polar to binomial form?
4. How can you check the correctness of complex number simplifications?
5. How do you interpret the imaginary unit \(i\) in real-world contexts?

### Tip:
Always simplify complex fractions by multiplying by the conjugate of the denominator to eliminate imaginary numbers from the denominator.

Learn how to simplify the complex fraction \(\frac{1+i}{1-i}\) using conjugates and express it in binomial form. This problem involves advanced concepts in complex numbers and algebraic manipulation.

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