distribute terms the collect real and imaginary parts (-m Power[\omega,2] + Power[k,2] + i\omega\gamma)(Subscript[x,-]cos(\phi) - i Subscript[x,-]sin(\phi)+ Subscript[x,+]cos(\phi)+Subscript[x,+] i sin(\phi))

To distribute the terms and then collect the real and imaginary parts of the expression

\[
(-m \omega^2 + k^2 + i\omega\gamma) \left( x_- \cos(\phi) - i x_- \sin(\phi) + x_+ \cos(\phi) + i x_+ \sin(\phi) \right),
\]

let's proceed step by step.

### Step 1: Expand the expression

Expand the product of the two terms:

\[
(-m \omega^2 + k^2 + i\omega\gamma) \cdot \left( x_- \cos(\phi) - i x_- \sin(\phi) + x_+ \cos(\phi) + i x_+ \sin(\phi) \right).
\]

Distribute each term in the first factor to each term in the second factor:

1. Distribute \(-m \omega^2\):

\[
-m \omega^2 \left( x_- \cos(\phi) - i x_- \sin(\phi) + x_+ \cos(\phi) + i x_+ \sin(\phi) \right) = -m \omega^2 x_- \cos(\phi) + i m \omega^2 x_- \sin(\phi) - m \omega^2 x_+ \cos(\phi) - i m \omega^2 x_+ \sin(\phi).
\]

2. Distribute \(k^2\):

\[
k^2 \left( x_- \cos(\phi) - i x_- \sin(\phi) + x_+ \cos(\phi) + i x_+ \sin(\phi) \right) = k^2 x_- \cos(\phi) - i k^2 x_- \sin(\phi) + k^2 x_+ \cos(\phi) + i k^2 x_+ \sin(\phi).
\]

3. Distribute \(i\omega\gamma\):

\[
i\omega\gamma \left( x_- \cos(\phi) - i x_- \sin(\phi) + x_+ \cos(\phi) + i x_+ \sin(\phi) \right) = i \omega \gamma x_- \cos(\phi) + \omega \gamma x_- \sin(\phi) + i \omega \gamma x_+ \cos(\phi) - \omega \gamma x_+ \sin(\phi).
\]

### Step 2: Combine like terms

Now, let's combine the terms with like real and imaginary parts:

#### Real Part:

- From \(-m \omega^2 x_- \cos(\phi)\)
- From \(- m \omega^2 x_+ \cos(\phi)\)
- From \(k^2 x_- \cos(\phi)\)
- From \(k^2 x_+ \cos(\phi)\)
- From \(\omega \gamma x_- \sin(\phi)\)
- From \(- \omega \gamma x_+ \sin(\phi)\)

Combining these:

\[
\text{Real part} = -m \omega^2 x_- \cos(\phi) - m \omega^2 x_+ \cos(\phi) + k^2 x_- \cos(\phi) + k^2 x_+ \cos(\phi) + \omega \gamma x_- \sin(\phi) - \omega \gamma x_+ \sin(\phi)
\]

\[
= \cos(\phi) \left[ (-m \omega^2 + k^2) (x_- + x_+) \right] + \sin(\phi) \left[ \omega \gamma (x_- - x_+) \right].
\]

#### Imaginary Part:

- From \(i m \omega^2 x_- \sin(\phi)\)
- From \(- i m \omega^2 x_+ \sin(\phi)\)
- From \(- i k^2 x_- \sin(\phi)\)
- From \(i k^2 x_+ \sin(\phi)\)
- From \(i \omega \gamma x_- \cos(\phi)\)
- From \(i \omega \gamma x_+ \cos(\phi)\)

Combining these:

\[
\text{Imaginary part} = i \left[ m \omega^2 x_- \sin(\phi) - m \omega^2 x_+ \sin(\phi) - k^2 x_- \sin(\phi) + k^2 x_+ \sin(\phi) + \omega \gamma x_- \cos(\phi) + \omega \gamma x_+ \cos(\phi) \right]
\]

\[
= i \left[ \sin(\phi) \left( (m \omega^2 - k^2) (x_- - x_+) \right) + \cos(\phi) \left( \omega \gamma (x_- + x_+) \right) \right].
\]

### Final Answer

The real part is:

\[
\cos(\phi) \left[ (-m \omega^2 + k^2) (x_- + x_+) \right] + \sin(\phi) \left[ \omega \gamma (x_- - x_+) \right].
\]

The imaginary part is:

\[
i \left[ \sin(\phi) \left( (m \omega^2 - k^2) (x_- - x_+) \right) + \cos(\phi) \left( \omega \gamma (x_- + x_+) \right) \right].
\]

Would you like more details or have any questions?

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**Related Questions:**
1. How does the distribution of terms work in complex numbers?
2. What is the significance of the real and imaginary parts in complex expressions?
3. Can this expression be simplified further if \(\phi\) is a specific angle?
4. How do you apply Euler's formula to convert trigonometric forms to exponential forms?
5. What happens to the expression if \(x_-\) and \(x_+\) are equal?

**Tip:** When working with complex expressions, always organize real and imaginary components separately to avoid errors in computation.

Explore how to distribute and collect real and imaginary parts of a complex expression involving trigonometric functions and complex coefficients. Step-by-step solution provided with detailed explanations of each mathematical operation.

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