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To expand \((x^2 - 5x + 6)^4\) using the **Multinomial Theorem**, we need to use the general formula for expanding powers of polynomials.

The **Multinomial Theorem** states that for a polynomial \((x_1 + x_2 + \cdots + x_m)^n\), the expansion is given by:

\[
\sum_{k_1+k_2+\cdots+k_m=n} \binom{n}{k_1, k_2, \ldots, k_m} \prod_{i=1}^{m} x_i^{k_i}
\]

where \(\binom{n}{k_1, k_2, \ldots, k_m}\) is the multinomial coefficient.

### Step-by-Step Process for \((x^2 - 5x + 6)^4\):

1. **Identify the terms:**
   - \(x_1 = x^2\),
   - \(x_2 = -5x\),
   - \(x_3 = 6\),
   and the exponent \(n = 4\).

2. **Find the multinomial coefficients:**
   We need to expand based on different combinations of powers \(k_1, k_2, k_3\) such that \(k_1 + k_2 + k_3 = 4\).

3. **Multinomial coefficients** for each combination:
   The multinomial coefficients are calculated using the formula:
   \[
   \binom{4}{k_1, k_2, k_3} = \frac{4!}{k_1! k_2! k_3!}
   \]
   We need to apply this for each set of exponents \(k_1, k_2, k_3\).

4. **Expand each term:**
   Each term in the expansion is of the form:
   \[
   \binom{4}{k_1, k_2, k_3} \cdot (x^2)^{k_1} \cdot (-5x)^{k_2} \cdot 6^{k_3}
   \]
   and we sum all the terms corresponding to each valid combination.

Let me calculate this expansion for you.The expansion of \((x^2 - 5x + 6)^4\) using the **Multinomial Theorem** is:

\[
x^8 - 20x^7 + 174x^6 - 860x^5 + 2641x^4 - 5160x^3 + 6264x^2 - 4320x + 1296
\]

Would you like further details on the calculation steps or explanations?

Here are five related questions to explore further:
1. How do you calculate multinomial coefficients for higher powers?
2. What is the significance of each term in the expansion?
3. Can we apply the Binomial Theorem for simpler polynomials?
4. How does the Multinomial Theorem generalize the Binomial Theorem?
5. Can this method be applied to any degree of polynomials?

**Tip:** The Multinomial Theorem can be applied to any polynomial raised to any power, as long as you carefully manage the sum of the exponents in each term.

Expand (x^2 - 5x + 6)^4 through Multinomial Theorem

Learn how to expand (x^2 - 5x + 6)^4 using the Multinomial Theorem with a detailed explanation and step-by-step calculation. This example covers advanced algebra concepts, suitable for high school or college-level students.

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