(\sum_{i=4}^{6}(3-2x_{i}))^{2}-2\sum_{i=1}^{6}6x_{i}

Let's break down the expression step by step and solve it.

The given expression is:

\[
\left(\sum_{i=4}^{6} (3 - 2x_i)\right)^2 - 2\sum_{i=1}^{6} 6x_i
\]

### Step 1: Evaluate the summation \(\sum_{i=4}^{6} (3 - 2x_i)\)
This summation expands as:

\[
\sum_{i=4}^{6} (3 - 2x_i) = (3 - 2x_4) + (3 - 2x_5) + (3 - 2x_6)
\]

So,

\[
\sum_{i=4}^{6} (3 - 2x_i) = 9 - 2(x_4 + x_5 + x_6)
\]

### Step 2: Square the result

\[
\left(\sum_{i=4}^{6} (3 - 2x_i)\right)^2 = \left(9 - 2(x_4 + x_5 + x_6)\right)^2
\]

### Step 3: Evaluate the summation \(\sum_{i=1}^{6} 6x_i\)

This summation expands as:

\[
\sum_{i=1}^{6} 6x_i = 6(x_1 + x_2 + x_3 + x_4 + x_5 + x_6)
\]

### Step 4: Multiply the result by \(-2\)

\[
-2\sum_{i=1}^{6} 6x_i = -12(x_1 + x_2 + x_3 + x_4 + x_5 + x_6)
\]

### Step 5: Combine all parts

The full expression now becomes:

\[
\left(9 - 2(x_4 + x_5 + x_6)\right)^2 - 12(x_1 + x_2 + x_3 + x_4 + x_5 + x_6)
\]

This is the simplified form of the expression.

Would you like more details or further steps?

### Related Questions:
1. How do you expand and simplify a squared binomial like \((a-b)^2\)?
2. How would this expression change if the summation limits were different?
3. What if there were additional terms inside the summation? How would it affect the final result?
4. Can you factorize the expression \(\left(9 - 2(x_4 + x_5 + x_6)\right)^2\)?
5. How does the expression simplify if \(x_i\) are specific numbers?

### Tip:
When working with summations, always ensure to evaluate them carefully before moving on to further operations like squaring or multiplication.

Learn how to simplify and solve a complex mathematical expression involving summations and squares step-by-step. This problem covers summation evaluation, binomial expansion, and quadratic equation concepts. Suitable for advanced high school students and beyond.

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