expand  2*alpha*p(2*k^2*p*(k*(2*a*gamma - 1)*alpha/2 - gamma^2*v/2)*(omega[o] + omega[1]*phi(Zeta))*omega[1]*phi(Zeta)*(2*phi(Zeta)^2 + a) - 2*k^2*(p - 1)*(k*(2*a*gamma - 1)*alpha/2 - gamma^2*v/2)*omega[1]^2*(rho + a*phi(Zeta)^2 + phi(Zeta)^4) + p^2*(a^2*k*(2*a*gamma + 3)*alpha + k*tau - v*(a*gamma + 1)^2)*(omega[o] + omega[1]*phi(Zeta))^2)

Let's expand the expression \( 2 \alpha p \) times the entire expression within the parentheses. I'll break down the expansion step by step:

### Expression:
\[
2\alpha p \left( 2k^2 p \left( k\left(\frac{(2a\gamma - 1)\alpha}{2}\right) - \frac{\gamma^2 v}{2} \right)(\omega[o] + \omega[1]\phi(\zeta))\omega[1]\phi(\zeta)\left(2\phi(\zeta)^2 + a\right) - 2k^2(p - 1)\left( k\left(\frac{(2a\gamma - 1)\alpha}{2}\right) - \frac{\gamma^2 v}{2} \right)\omega[1]^2 \left(\rho + a\phi(\zeta)^2 + \phi(\zeta)^4 \right) + p^2\left( a^2 k(2a\gamma + 3)\alpha + k\tau - v(a\gamma + 1)^2 \right) (\omega[o] + \omega[1]\phi(\zeta))^2 \right)
\]

### Step 1: Distribute \( 2 \alpha p \) across the three terms inside the parentheses.

#### First Term:
\[
2 \alpha p \cdot \left[ 2k^2 p \left( k\left(\frac{(2a\gamma - 1)\alpha}{2}\right) - \frac{\gamma^2 v}{2} \right)(\omega[o] + \omega[1]\phi(\zeta))\omega[1]\phi(\zeta)\left(2\phi(\zeta)^2 + a\right) \right]
\]

\[
= 4 \alpha p k^2 p \left( k\left(\frac{(2a\gamma - 1)\alpha}{2}\right) - \frac{\gamma^2 v}{2} \right)(\omega[o] + \omega[1]\phi(\zeta))\omega[1]\phi(\zeta)\left(2\phi(\zeta)^2 + a\right)
\]

#### Second Term:
\[
2 \alpha p \cdot \left[ -2k^2(p - 1)\left( k\left(\frac{(2a\gamma - 1)\alpha}{2}\right) - \frac{\gamma^2 v}{2} \right)\omega[1]^2 \left(\rho + a\phi(\zeta)^2 + \phi(\zeta)^4 \right) \right]
\]

\[
= -4 \alpha p k^2(p - 1)\left( k\left(\frac{(2a\gamma - 1)\alpha}{2}\right) - \frac{\gamma^2 v}{2} \right)\omega[1]^2 \left(\rho + a\phi(\zeta)^2 + \phi(\zeta)^4 \right)
\]

#### Third Term:
\[
2 \alpha p \cdot \left[ p^2\left( a^2 k(2a\gamma + 3)\alpha + k\tau - v(a\gamma + 1)^2 \right) (\omega[o] + \omega[1]\phi(\zeta))^2 \right]
\]

\[
= 2 \alpha p^3\left( a^2 k(2a\gamma + 3)\alpha + k\tau - v(a\gamma + 1)^2 \right) (\omega[o] + \omega[1]\phi(\zeta))^2
\]

### Final Expanded Expression:
Putting all three expanded terms together, the expression becomes:

\[
4 \alpha p k^2 p \left( k\left(\frac{(2a\gamma - 1)\alpha}{2}\right) - \frac{\gamma^2 v}{2} \right)(\omega[o] + \omega[1]\phi(\zeta))\omega[1]\phi(\zeta)\left(2\phi(\zeta)^2 + a\right)
\]

\[
- 4 \alpha p k^2(p - 1)\left( k\left(\frac{(2a\gamma - 1)\alpha}{2}\right) - \frac{\gamma^2 v}{2} \right)\omega[1]^2 \left(\rho + a\phi(\zeta)^2 + \phi(\zeta)^4 \right)
\]

\[
+ 2 \alpha p^3\left( a^2 k(2a\gamma + 3)\alpha + k\tau - v(a\gamma + 1)^2 \right) (\omega[o] + \omega[1]\phi(\zeta))^2
\]

This is the fully expanded form of the given expression.

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Would you like a detailed explanation of any specific part of this expansion, or do you have any questions? 

Here are five related questions you might find interesting:

1. How can this expansion be simplified further?
2. How does the expansion change if the variable \( \alpha \) is set to 1?
3. What are the implications of setting \( \phi(\zeta) \) to a specific value?
4. How does the expansion behave under the limit \( \gamma \to 0 \)?
5. Can this expression be factorized in any way?

**Tip:** When expanding complex algebraic expressions, distribute multiplication carefully and watch for common factors that can simplify the expression.

Learn how to expand a complex algebraic expression involving polynomials and multiple variables step-by-step. This detailed solution covers the distribution of terms and factors, suitable for advanced mathematics.

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