Let's break down and simplify the expression you provided step by step: 

\[
-2(-x)^5(6x-9)^4((-x)^2 + 1)^2
\]

### Step 1: Simplify each component

#### \( (-x)^5 \)
Since the exponent is odd, this expression simplifies as follows:

\[
(-x)^5 = -x^5
\]

#### \( (6x - 9)^4 \)
We have already expanded \( (6x - 9)^4 \) in the previous step. The result was:

\[
(6x - 9)^4 = 1296x^4 - 7776x^3 + 17496x^2 - 17496x + 6561
\]

#### \( (-x)^2 + 1 \)
Here, \( (-x)^2 \) simplifies to \( x^2 \), so:

\[
(-x)^2 + 1 = x^2 + 1
\]

Now, squaring this expression gives:

\[
(x^2 + 1)^2 = x^4 + 2x^2 + 1
\]

### Step 2: Combine the components
Now that we've simplified the individual components, we can substitute them back into the original expression:

\[
-2(-x)^5(6x - 9)^4((-x)^2 + 1)^2
\]

Substituting the simplifications:

\[
-2(-x^5)(1296x^4 - 7776x^3 + 17496x^2 - 17496x + 6561)(x^4 + 2x^2 + 1)
\]

### Step 3: Simplify the negative terms
We already have \( -x^5 \) inside the expression. Multiplying by the negative 2 will simplify as:

\[
-2(-x^5) = 2x^5
\]

So the expression becomes:

\[
2

In this lesson, we simplify the complex algebraic expression -2(-x)^5(6x-9)^4((-x)^2+1)^2 step-by-step, covering polynomial expansion, exponent rules, and binomial expressions. This solution is suitable for advanced high school students (Grades 10-12).

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